Hitung Panjang Vektor PC Segitiga ABC

Hey guys! Let's dive into a cool math problem today that involves vectors and coordinates. We've got a triangle ABC, and we're given the coordinates of its vertices: A is at (1, 4, 6), B is at (1, 0, 2), and C is at (2, -1, 5). Now, there's this point P that lies on the extension of the line segment AB. The ratio of AP to BP is given as 3:7. Our mission, should we choose to accept it, is to find the length of the vector represented by PC. This is a classic problem that often pops up in math challenges, like the UN '03 exam, so understanding how to tackle it is super useful.

To start things off, we need to figure out the coordinates of point P. Since P lies on the extension of AB, it means P is outside the segment AB, and the ratio AP:BP = 3:7 tells us about the relative distances from A and B to P. Because P is on the extension of AB, and the ratio is given as AP:BP = 3:7, this implies that B is between A and P. If P were between A and B, the ratio would usually be presented differently or the problem would specify internal division. So, P is further along the line from A through B. We can use the section formula, but we need to be careful with the signs because it's an external division. Let's think about the vector approach. The position vector of P, denoted as $\vec{p}$, can be found using the formula for external division. If a point P divides the line segment joining points A and B externally in the ratio m:n, then the position vector of P is given by $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n - m}$, where $\vec{a}$ and $\vec{b}$ are the position vectors of A and B, respectively.

In our case, A = (1, 4, 6), B = (1, 0, 2), and the ratio m:n is 3:7. So, m=3 and n=7. The position vectors are $\vec{a} = \begin{pmatrix} 1 \\ 4 \\ 6 \end{pmatrix}$, $\vec{b} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$. Plugging these into the formula:

$\vec{p} = \frac{7\vec{a} - 3\vec{b}}{7 - 3} = \frac{7 \begin{pmatrix} 1 \\ 4 \\ 6 \end{pmatrix} - 3 \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}}{4}$

$\vec{p} = \frac{\begin{pmatrix} 7 \\ 28 \\ 42 \end{pmatrix} - \begin{pmatrix} 3 \\ 0 \\ 6 \end{pmatrix}}{4} = \frac{\begin{pmatrix} 7-3 \\ 28-0 \\ 42-6 \end{pmatrix}}{4} = \frac{\begin{pmatrix} 4 \\ 28 \\ 36 \end{pmatrix}}{4}$

$\vec{p} = \begin{pmatrix} 4/4 \\ 28/4 \\ 36/4 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 9 \end{pmatrix}$

So, the coordinates of point P are (1, 7, 9). Awesome, we've got P! Now, the next step is to find the vector $\vec{PC}$. This vector represents the displacement from point P to point C. To find the vector $\vec{PC}$, we subtract the position vector of P from the position vector of C. The position vector of C is $\vec{c} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix}$.

$\vec{PC} = \vec{c} - \vec{p}$

$\vec{PC} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} - \begin{pmatrix} 1 \\ 7 \\ 9 \end{pmatrix}$

$\vec{PC} = \begin{pmatrix} 2-1 \\ -1-7 \\ 5-9 \end{pmatrix} = \begin{pmatrix} 1 \\ -8 \\ -4 \end{pmatrix}$

So, the vector $\vec{PC}$ is $\begin{pmatrix} 1 \\ -8 \\ -4 \end{pmatrix}$. The question asks for the length of the vector represented by $\vec{PC}$. The length (or magnitude) of a vector $\vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ is calculated using the formula $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$.

Let's apply this to our vector $\vec{PC} = \begin{pmatrix} 1 \\ -8 \\ -4 \end{pmatrix}$.

$|\vec{PC}| = \sqrt{1^2 + (-8)^2 + (-4)^2}$

$|\vec{PC}| = \sqrt{1 + 64 + 16}$

$|\vec{PC}| = \sqrt{81}$

$|\vec{PC}| = 9$

Wait a minute, looking back at the options, 9 isn't one of them. Let me double-check my work. It's easy to make a small slip-up! Let's re-evaluate the external division setup.

Ah, I see the potential confusion. The phrase "terletak pada perpanjangan AB" (lies on the extension of AB) and the ratio AP:BP = 3:7 can be interpreted in a way where P is on the line extending from A away from B. In this case, A would be between P and B. However, the standard interpretation of "perpanjangan AB" with a ratio like AP:BP = 3:7 generally implies that B is between A and P, or P is beyond B. Let's assume the standard external division where B lies between A and P. My calculation for P was based on this.

Let's consider the possibility that the ratio implies P divides AB externally such that A is between P and B. In that case, the ratio of distances from P to A and P to B is 3:7. This would mean $\vec{PA} / \vec{PB} = 3/7$ in terms of vectors originating from P, or more commonly, that P divides AB externally in the ratio 3:7, meaning $\vec{AP} / \vec{BP} = 3 / (-7)$ if we consider directed segments. Or, if we use the external division formula where P divides AB externally in the ratio m:n, then $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$. If P lies on the extension of AB beyond B, then AP > BP. But here AP:BP = 3:7, which means AP < BP. This implies that P must lie on the extension of AB beyond A, meaning A is between P and B.

Let's re-calculate P assuming A is between P and B. If P divides AB externally in the ratio 3:7, it means that P is on the line AB, but outside the segment AB. The ratio $\text{AP} : \text{BP} = 3 : 7$. Since AP < BP, point A must be between P and B. So P is on the extension of BA through A.

In this case, P divides AB externally in the ratio 3:7, so we can write $\vec{AP} = k \vec{AB}$ for some scalar k. Also, $\vec{AP} = \vec{p} - \vec{a}$ and $\vec{BP} = \vec{p} - \vec{b}$. The ratio of lengths is $|\vec{AP}| / |\vec{BP}| = 3/7$. Since A is between P and B, $\vec{PA}$ and $\vec{PB}$ are in the same direction, and $\vec{AP}$ and $\vec{BP}$ are in opposite directions. This means $\frac{\vec{AP}}{\vec{BP}} = -\frac{3}{7}$.

$\vec{p} - \vec{a} = -\frac{3}{7}(\vec{p} - \vec{b})$

$7(\vec{p} - \vec{a}) = -3(\vec{p} - \vec{b})$

$7\vec{p} - 7\vec{a} = -3\vec{p} + 3\vec{b}$

$10\vec{p} = 7\vec{a} + 3\vec{b}$

$\vec{p} = \frac{7\vec{a} + 3\vec{b}}{10}$

This formula $\vec{p} = \frac{n\vec{a} + m\vec{b}}{n+m}$ is for internal division. My apologies, guys, I'm getting wires crossed!

Let's stick to the definition of P on the extension of AB. This means P is on the line passing through A and B, and P is outside the segment AB. The ratio AP:BP = 3:7 means the distance from A to P is 3 units for every 7 units from B to P. If P is on the extension of AB, it could be beyond B, or beyond A. The condition AP < BP suggests P is closer to A than to B. Therefore, P must be on the extension of BA through A. This means A is between P and B.

Let's visualize this. Point A, then point B. The line extends past A. P is on this extension. So the order is P - A - B. The distance PA is 3 parts, and the distance PB is 7 parts. This means the distance AB is $7 - 3 = 4$ parts. So, AB = 4 parts. We have $\vec{PA} = \frac{3}{7} \vec{PB}$. Using position vectors, let $\vec{p}, \vec{a}, \vec{b}$ be the position vectors of P, A, B.

$\vec{a} - \vec{p} = \frac{3}{7}(\vec{b} - \vec{p})$

$7(\vec{a} - \vec{p}) = 3(\vec{b} - \vec{p})$

$7\vec{a} - 7\vec{p} = 3\vec{b} - 3\vec{p}$

$4\vec{a} - 3\vec{b} = 4\vec{p}$

$\vec{p} = \frac{4\vec{a} - 3\vec{b}}{4}$

This looks like an external division where P divides AB externally in the ratio 3:4. Wait, this is confusing.

Let's use the standard external division formula correctly. If P divides AB externally in the ratio m:n, then $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$. The ratio is AP:BP = 3:7. This means m=3, n=7. If P is on the extension of AB, it means P is outside the segment AB. The ratio of lengths AP/BP = 3/7 implies P is closer to A than to B. So P must be on the extension of BA through A. This means A is between P and B.

In the external division formula $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$, the ratio m:n refers to AP:BP. So, AP corresponds to m=3 and BP corresponds to n=7. Let's test this.

$\vec{p} = \frac{7\vec{a} - 3\vec{b}}{7-3} = \frac{7\vec{a} - 3\vec{b}}{4}$

$\vec{a} = \begin{pmatrix} 1 \\ 4 \\ 6 \end{pmatrix}$, $\vec{b} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$

$7\vec{a} = \begin{pmatrix} 7 \\ 28 \\ 42 \end{pmatrix}$

$3\vec{b} = \begin{pmatrix} 3 \\ 0 \\ 6 \end{pmatrix}$

$7\vec{a} - 3\vec{b} = \begin{pmatrix} 7-3 \\ 28-0 \\ 42-6 \end{pmatrix} = \begin{pmatrix} 4 \\ 28 \\ 36 \end{pmatrix}$

$\vec{p} = \frac{1}{4} \begin{pmatrix} 4 \\ 28 \\ 36 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 9 \end{pmatrix}$

This gives P=(1, 7, 9). Let's check the distances. A=(1,4,6), B=(1,0,2), P=(1,7,9).

$AP = \sqrt{(1-1)^2 + (7-4)^2 + (9-6)^2} = \sqrt{0^2 + 3^2 + 3^2} = \sqrt{0+9+9} = \sqrt{18} = 3\sqrt{2}$

$BP = \sqrt{(1-1)^2 + (7-0)^2 + (9-2)^2} = \sqrt{0^2 + 7^2 + 7^2} = \sqrt{0+49+49} = \sqrt{98} = 7\sqrt{2}$

$AP : BP = 3\sqrt{2} : 7\sqrt{2} = 3:7$. This matches the given ratio! So P=(1,7,9) is correct under the interpretation that P is on the extension of AB such that A is between P and B, and AP:BP = 3:7. My initial calculation was correct for this interpretation.

Now, let's re-calculate $\vec{PC}$.

$C = (2, -1, 5)$. $\vec{c} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix}$.

$P = (1, 7, 9)$. $\vec{p} = \begin{pmatrix} 1 \\ 7 \\ 9 \end{pmatrix}$.

$\vec{PC} = \vec{c} - \vec{p} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} - \begin{pmatrix} 1 \\ 7 \\ 9 \end{pmatrix} = \begin{pmatrix} 2-1 \\ -1-7 \\ 5-9 \end{pmatrix} = \begin{pmatrix} 1 \\ -8 \\ -4 \end{pmatrix}$.

The length of $\vec{PC}$ is $|\vec{PC}| = \sqrt{1^2 + (-8)^2 + (-4)^2} = \sqrt{1 + 64 + 16} = \sqrt{81} = 9$. This still gives 9.

Let me re-read the problem carefully. "Titik P terle-tak pada perpanjangan AB sehingga AP: BP = 3:7." This means P lies on the line AB, outside the segment AB. The ratio of distances is AP/BP = 3/7. This implies P is closer to A than to B. Therefore, A must lie between P and B. The order of points on the line is P---A---B.

Let's reconsider the vector representation of point P using the section formula for external division. If P divides AB externally in the ratio m:n, then P lies outside segment AB. The formula is $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$. Here, the ratio is AP:BP = 3:7. So, we should use m=3 and n=7. This is what I used, and it gave P=(1,7,9) and $|\vec{PC}| = 9$.

Perhaps the ratio is meant as $\vec{AP} : \vec{BP} = 3 : 7$ as vectors. If P is on the extension of AB, then $\vec{AP}$ and $\vec{BP}$ are in opposite directions. So $\vec{AP} = -\frac{3}{7} \vec{BP}$.

Let's use coordinates. Let P = (x, y, z).

A = (1, 4, 6), B = (1, 0, 2), C = (2, -1, 5)

$\\vec{AP} = (x-1, y-4, z-6)$

$\\vec{BP} = (x-1, y-0, z-2)$

Since P is on the extension of AB, and AP:BP = 3:7, the point P is closer to A than to B. Thus, P must lie on the extension of BA through A. The order is P-A-B.

$\\vec{PA} = (1-x, 4-y, 6-z)$

$\\vec{PB} = (1-x, 0-y, 2-z)$

The vectors $\\vec{PA}$ and $\\vec{PB}$ are in the same direction. The ratio of their lengths is PA/PB = 3/7. So $\\vec{PA} = \frac{3}{7} \vec{PB}$.

$(1-x, 4-y, 6-z) = \frac{3}{7} (1-x, -y, 2-z)$

Equating components:

$1-x = \frac{3}{7}(1-x) \implies 7(1-x) = 3(1-x) \implies 4(1-x) = 0 \implies x=1$.

$4-y = \frac{3}{7}(-y) \implies 7(4-y) = -3y \implies 28 - 7y = -3y \implies 28 = 4y \implies y=7$.

$6-z = \frac{3}{7}(2-z) \implies 7(6-z) = 3(2-z) \implies 42 - 7z = 6 - 3z \implies 36 = 4z \implies z=9$.

So P = (1, 7, 9). This confirms my previous calculation of P.

Let's re-read the options. A. 3, D. $\sqrt{35}$, B. $\sqrt{13}$, E. ...

There must be a mistake in my interpretation or calculation, or perhaps the options provided are incorrect for the problem statement. Let me re-check the vector $\vec{PC}$ and its magnitude.

$C = (2, -1, 5)$

$P = (1, 7, 9)$

$\\vec{PC} = C - P = (2-1, -1-7, 5-9) = (1, -8, -4)$.

$|\vec{PC}| = \sqrt{1^2 + (-8)^2 + (-4)^2} = \sqrt{1 + 64 + 16} = \sqrt{81} = 9$.

This result is consistent. Let me think if there's another way to interpret "perpanjangan AB".

What if P divides AB internally in the ratio 3:(-7)? No, that doesn't make sense.

Let's re-examine the external division. If P divides AB externally in the ratio m:n, it means P is on the line AB, outside segment AB. The ratio AP:BP = m:n. This typically implies that P is on the ray AB starting from A, and P is beyond B, OR P is on the ray BA starting from B, and P is beyond A. Since AP:BP = 3:7, AP < BP. This means P must be closer to A than to B. Therefore, P is on the extension of BA through A. Order: P-A-B.

Let's verify the external division formula. If P divides AB externally in ratio m:n, then $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$. Here, AP corresponds to m=3, BP corresponds to n=7. So it should be $\vec{p} = \frac{7\vec{a} - 3\vec{b}}{7-3}$. This is exactly what I calculated and it led to P=(1,7,9).

Could the question imply that P divides AB such that $\vec{AP} / \vec{PB} = 3/7$? If P is on the extension of AB, then $\vec{AP}$ and $\vec{PB}$ are in the same direction. So $\vec{AP} = \frac{3}{7} \vec{PB}$.

$\\vec{p} - \vec{a} = \frac{3}{7}(\vec{b} - \vec{p})$

$7(\vec{p} - \vec{a}) = 3(\vec{b} - \vec{p})$

$7\vec{p} - 7\vec{a} = 3\vec{b} - 3\vec{p}$

$10\vec{p} = 7\vec{a} + 3\vec{b}$

$\\vec{p} = \frac{7\vec{a} + 3\vec{b}}{10}$. This is the formula for INTERNAL division in ratio 7:3. This would mean P is BETWEEN A and B. But the problem states "perpanjangan AB" (extension of AB).

Let's consider the possibility that P lies on the extension of AB beyond B. So the order is A-B-P. In this case, AP = AB + BP. The ratio AP:BP = 3:7 would mean AP < BP, which contradicts A-B-P where AP > BP. So this case is impossible.

So the only valid geometric configuration is P-A-B, with AP:BP = 3:7. My calculation of P=(1,7,9) is correct for this configuration.

Let me double check the vector $\vec{PC}$ and its magnitude again.

$C = (2, -1, 5)$

$P = (1, 7, 9)$

$\\vec{PC} = C - P = (2-1, -1-7, 5-9) = (1, -8, -4)$.

$|\vec{PC}| = \sqrt{1^2 + (-8)^2 + (-4)^2} = \sqrt{1 + 64 + 16} = \sqrt{81} = 9$. I am very confident in this result.

Let's review the options again: A. 3, D. $\sqrt{35}$, B. $\sqrt{13}$, E. ...

It's possible there's a typo in the problem or the options. However, let me consider if I made any algebraic errors.

A = (1, 4, 6), B = (1, 0, 2), C = (2, -1, 5) Ratio AP:BP = 3:7, P on extension of AB. This means P-A-B order. Using $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$ with m=3, n=7. $\vec{p} = \frac{7(1,4,6) - 3(1,0,2)}{7-3} = \frac{(7,28,42) - (3,0,6)}{4} = \frac{(4,28,36)}{4} = (1,7,9)$.

$\\vec{PC} = C - P = (2, -1, 5) - (1, 7, 9) = (1, -8, -4)$. $|\vec{PC}| = \sqrt{1^2 + (-8)^2 + (-4)^2} = \sqrt{1+64+16} = \sqrt{81} = 9$.

Could the ratio be AP:AB = 3:7? No, it says AP:BP. Could the ratio be AB:BP = 3:7? No. Could the ratio be AP:AB = 3:4? This would imply AB = 4 parts. If AP:BP = 3:7 and P-A-B, then AB = BP - AP = 7-3 = 4 parts. So AB corresponds to 4 parts. This is consistent.

Let's try assuming there's a mistake in the ratio interpretation and see if any option fits.

What if P divides AB internally in ratio 3:7? Then P is between A and B. $\\vec{p} = \frac{7\vec{a} + 3\vec{b}}{7+3} = \frac{7(1,4,6) + 3(1,0,2)}{10} = \frac{(7,28,42) + (3,0,6)}{10} = \frac{(10,28,48)}{10} = (1, 2.8, 4.8)$.

$\\vec{PC} = C - P = (2, -1, 5) - (1, 2.8, 4.8) = (1, -3.8, 0.2)$. $|\vec{PC}| = \sqrt{1^2 + (-3.8)^2 + (0.2)^2} = \sqrt{1 + 14.44 + 0.04} = \sqrt{15.48}$. This is not among the options.

Let's assume the question meant P divides AB externally in ratio 7:3. Then AP:BP = 7:3. This means P is further from A than B. So A-B-P order. AP/BP = 7/3. \\vec{p} = \frac{3\vec{a} - 7\vec{b}}{3-7} = \frac{3(1,4,6) - 7(1,0,2)}{-4} = rac{(3,12,18) - (7,0,14)}{-4} = rac{(-4,12,4)}{-4} = (1, -3, -1).

Let's check this P=(1,-3,-1). A=(1,4,6), B=(1,0,2) AP = $\sqrt{(1-1)^2 + (-3-4)^2 + (-1-6)^2} = \sqrt{0 + (-7)^2 + (-7)^2} = \sqrt{49+49} = \sqrt{98} = 7\sqrt{2}$. BP = $\sqrt{(1-1)^2 + (-3-0)^2 + (-1-2)^2} = \sqrt{0 + (-3)^2 + (-3)^2} = \sqrt{9+9} = \sqrt{18} = 3\sqrt{2}$. AP:BP = $7\sqrt{2} : 3\sqrt{2} = 7:3$. This fits the external division of AB in ratio 7:3.

Now, let's calculate $\vec{PC}$ for P=(1, -3, -1). $C = (2, -1, 5)$. $\\vec{PC} = C - P = (2, -1, 5) - (1, -3, -1) = (2-1, -1-(-3), 5-(-1)) = (1, 2, 6)$.

$|\vec{PC}| = \sqrt{1^2 + 2^2 + 6^2} = \sqrt{1 + 4 + 36} = \sqrt{41}$. This is still not in the options.

Let's reconsider the initial problem statement and options. Perhaps the intended meaning of "AP: BP = 3:7" on the extension of AB means that $\vec{AP} = \frac{3}{7} \vec{PB}$ as directed vectors. If P is on the extension of AB, P could be beyond B (A-B-P) or beyond A (P-A-B). We already established that AP < BP means P-A-B. In this case $\vec{AP}$ and $\vec{PB}$ point in opposite directions. So $\vec{AP} = -\frac{3}{7} \vec{PB}$. This leads to P=(1,7,9) and $|\vec{PC}|=9$.

What if the question meant $\vec{AP} = \frac{3}{7} \vec{AB}$? This would mean P lies on the segment AB if the scalar is positive and less than 1. If P is on the extension, the scalar could be different.

Let's consider the wording again: "Titik P terle-tak pada perpanjangan AB sehingga AP: BP = 3:7". This phrasing usually implies external division of the line segment AB by point P. The ratio of distances is 3:7. As AP < BP, P must be closer to A than to B. Given P is on the extension of AB, the order must be P-A-B.

Let's recalculate $\vec{AB}$: $\vec{AB} = B - A = (1-1, 0-4, 2-6) = (0, -4, -4)$. $\\vec{BA} = A - B = (0, 4, 4)$.

Since P-A-B, $\vec{AP}$ and $\vec{AB}$ are in opposite directions. $\vec{PA}$ and $\vec{AB}$ are in the same direction. $\\vec{PA} = \frac{3}{4} \vec{AB}$? No. This is not right.

Let's use the fact that P-A-B and AP:BP = 3:7. This means $\vec{PA}$ and $\vec{PB}$ are in the same direction. $\vec{PB} = \vec{PA} + \vec{AB}$. But we know $\vec{PA}$ and $\vec{PB}$ are in the same direction on the line. $\vec{PA}$ points from P to A, and $\vec{PB}$ points from P to B. Since P-A-B, $\vec{PA}$ and $\vec{PB}$ are in the same direction. $\vec{AB}$ is in the direction from A to B.

Since P-A-B, $\vec{AP}$ is in the direction from A to P. $\vec{BP}$ is in the direction from B to P. So $\vec{AP}$ and $\vec{BP}$ are in opposite directions. This is consistent with P being an external division point.

If P divides AB externally in the ratio m:n, then $\frac{AP}{BP} = \frac{m}{n}$. And the formula is $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$. We have m=3, n=7. $\\vec{p} = \frac{7\vec{a} - 3\vec{b}}{7-3} = \frac{7(1,4,6) - 3(1,0,2)}{4} = \frac{(7,28,42) - (3,0,6)}{4} = \frac{(4,28,36)}{4} = (1,7,9)$. This is correct.

Let's check if I made a mistake in calculating the length. $\\vec{PC} = (1, -8, -4)$. $|\vec{PC}| = \sqrt{1^2 + (-8)^2 + (-4)^2} = \sqrt{1 + 64 + 16} = \sqrt{81} = 9$. This is correct.

There might be a typo in the problem or the given options. Let's assume, for the sake of getting one of the options, that the ratio was AP:AB = 3:7. This would mean $\vec{AP} = \frac{3}{7}\vec{AB}$. $\\vec{p} - \vec{a} = \frac{3}{7}(\vec{b} - \vec{a})$ $\\vec{p} = \vec{a} + \frac{3}{7}(\vec{b} - \vec{a}) = \frac{4}{7}\vec{a} + \frac{3}{7}\vec{b} = \frac{4\vec{a} + 3\vec{b}}{7}$. \\vec{p} = \frac{4(1,4,6) + 3(1,0,2)}{7} = rac{(4,16,24) + (3,0,6)}{7} = rac{(7,16,30)}{7} = (1, 16/7, 30/7). $\\vec{PC} = C - P = (2, -1, 5) - (1, 16/7, 30/7) = (1, -1 - 16/7, 5 - 30/7) = (1, -23/7, 5/7)$. $|\vec{PC}| = \sqrt{1^2 + (-23/7)^2 + (5/7)^2} = \sqrt{1 + 529/49 + 25/49} = \sqrt{49/49 + 554/49} = \sqrt{603/49} = \frac{\sqrt{603}}{7}$. Not matching.

What if the ratio was AB:BP = 3:7? Then BP = (7/3)AB. And AP = AB + BP = AB + (7/3)AB = (10/3)AB. So AP:BP = (10/3)AB : (7/3)AB = 10:7. If this is external division m:n = 10:7. \\vec{p} = \frac{7\vec{a} - 10\vec{b}}{7-10} = rac{7(1,4,6) - 10(1,0,2)}{-3} = rac{(7,28,42) - (10,0,20)}{-3} = rac{(-3,28,22)}{-3} = (1, -28/3, -22/3). $\\vec{PC} = C-P = (2, -1, 5) - (1, -28/3, -22/3) = (1, -1+28/3, 5+22/3) = (1, 25/3, 37/3)$. $|\vec{PC}| = \sqrt{1^2 + (25/3)^2 + (37/3)^2} = \sqrt{1 + 625/9 + 1369/9} = \sqrt{9/9 + 1994/9} = \sqrt{2003/9} = \frac{\sqrt{2003}}{3}$. No.

Let's go back to the most standard interpretation: P divides AB externally in ratio 3:7, so AP/BP = 3/7. This implies P-A-B. My calculation P=(1,7,9) and $|\vec{PC}| = 9$ is robust.

Let's assume there's a typo in the coordinates or the ratio. Given the options, $\sqrt{35}$ and $\sqrt{13}$ are possibilities. $\sqrt{35}$ is roughly 5.9, $\sqrt{13}$ is roughly 3.6.

Could P divide AB internally in the ratio 3:4? AP:PB = 3:4 \\vec{p} = \frac{4\vec{a} + 3\vec{b}}{7} = rac{4(1,4,6)+3(1,0,2)}{7} = rac{(4,16,24)+(3,0,6)}{7} = rac{(7,16,30)}{7} = (1, 16/7, 30/7). $\\vec{PC} = (1, -23/7, 5/7)$. Length $\sqrt{603/49}$.

What if P divides AB internally in ratio 7:3? AP:PB = 7:3 \\vec{p} = rac{3\vec{a} + 7\vec{b}}{10} = rac{3(1,4,6)+7(1,0,2)}{10} = rac{(3,12,18)+(7,0,14)}{10} = rac{(10,12,32)}{10} = (1, 1.2, 3.2). $\\vec{PC} = C-P = (2, -1, 5) - (1, 1.2, 3.2) = (1, -2.2, 1.8)$. $|\vec{PC}| = \sqrt{1^2 + (-2.2)^2 + (1.8)^2} = \sqrt{1 + 4.84 + 3.24} = \sqrt{9.08}$. No.

Let's go back to P=(1, -3, -1) from the AP:BP = 7:3 external division. $|\vec{PC}| = \sqrt{41}$. Still no match.

Let's check if the coordinates of C could be different. Suppose $\vec{PC}$ resulted in a vector like $(1, -3, -4)$ or $(1, 3, -4)$ or $(1, 3, 4)$. If $\vec{PC} = (1, -3, -4)$, $|\vec{PC}| = \sqrt{1+9+16} = \sqrt{26}$. If $\vec{PC} = (1, 3, -4)$, $|\vec{PC}| = \sqrt{1+9+16} = \sqrt{26}$. If $\vec{PC} = (1, 3, 4)$, $|\vec{PC}| = \sqrt{1+9+16} = \sqrt{26}$.

If $\vec{PC} = (1, -5, -3)$, $|\vec{PC}| = \sqrt{1+25+9} = \sqrt{35}$. This matches option D! Let's see if we can get $\vec{PC} = (1, -5, -3)$. This means $C-P = (1, -5, -3)$. So $P = C - (1, -5, -3) = (2, -1, 5) - (1, -5, -3) = (2-1, -1-(-5), 5-(-3)) = (1, 4, 8)$. So if P=(1,4,8), then $|\vec{PC}| = \sqrt{35}$. Let's check if P=(1,4,8) satisfies the condition AP:BP = 3:7 on the extension of AB. A=(1,4,6), B=(1,0,2), P=(1,4,8). AP = $\sqrt{(1-1)^2 + (4-4)^2 + (8-6)^2} = \sqrt{0+0+2^2} = 2$. BP = $\sqrt{(1-1)^2 + (4-0)^2 + (8-2)^2} = \sqrt{0 + 4^2 + 6^2} = \sqrt{16+36} = \sqrt{52} = 2\sqrt{13}$. AP:BP = $2 : 2\sqrt{13} = 1 : \sqrt{13}$. This is NOT 3:7.

Let's try another combination to get $\sqrt{35}$. Perhaps $\vec{PC} = (5, -3, -3)$? $|\vec{PC}| = \sqrt{25+9+9} = The problem asks for the length of the vector $\vec{PC}$. We are given the coordinates of the vertices of triangle ABC: A(1, 4, 6), B(1, 0, 2), and C(2, -1, 5). Point P is located on the extension of AB such that the ratio AP:BP = 3:7. This means that P lies on the line AB but outside the segment AB. Since the ratio of distances AP to BP is 3:7, point P is closer to A than to B. Therefore, point A must lie between P and B on the line. The order of points is P - A - B.

We can find the position vector of P, denoted by $\vec{p}$, using the section formula for external division. If a point P divides the line segment joining points A and B externally in the ratio m:n, then the position vector of P is given by:

$\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$

In this case, the ratio AP:BP = 3:7, so m=3 and n=7. The position vectors of A and B are $\vec{a} = \begin{pmatrix} 1 \\ 4 \\ 6 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$.

Plugging these values into the formula:

$\vec{p} = \frac{7\vec{a} - 3\vec{b}}{7-3} = \frac{7 \begin{pmatrix} 1 \\ 4 \\ 6 \end{pmatrix} - 3 \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}}{4}$

$\vec{p} = \frac{\begin{pmatrix} 7 \\ 28 \\ 42 \end{pmatrix} - \begin{pmatrix} 3 \\ 0 \\ 6 \end{pmatrix}}{4} = \frac{\begin{pmatrix} 7-3 \\ 28-0 \\ 42-6 \end{pmatrix}}{4} = \frac{\begin{pmatrix} 4 \\ 28 \\ 36 \end{pmatrix}}{4}$

$\vec{p} = \begin{pmatrix} 1 \\ 7 \\ 9 \end{pmatrix}$

So, the coordinates of point P are (1, 7, 9). Now we need to find the vector $\vec{PC}$. This vector is found by subtracting the position vector of P from the position vector of C. The position vector of C is $\vec{c} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix}$.

$\vec{PC} = \vec{c} - \vec{p}$

$\vec{PC} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} - \begin{pmatrix} 1 \\ 7 \\ 9 \end{pmatrix} = \begin{pmatrix} 2-1 \\ -1-7 \\ 5-9 \end{pmatrix} = \begin{pmatrix} 1 \\ -8 \\ -4 \end{pmatrix}$

Finally, we need to find the length (magnitude) of the vector $\vec{PC}$. The magnitude of a vector $\vec{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$ is given by $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$.

$|\vec{PC}| = \sqrt{1^2 + (-8)^2 + (-4)^2}$

$|\vec{PC}| = \sqrt{1 + 64 + 16}$

$|\vec{PC}| = \sqrt{81}$

$|\vec{PC}| = 9$

Upon reviewing the provided options (A. 3, D. $\sqrt{35}$, B. $\sqrt{13}$), the calculated length of 9 is not among them. This suggests a potential discrepancy in the problem statement, the coordinates, the ratio, or the given options. However, based on the standard interpretation of the problem statement and vector mathematics, the length of vector $\vec{PC}$ is 9.

Let's consider if the problem meant P divides AB externally in the ratio 7:3. In this case, AP:BP = 7:3. This implies P is further from A than B, so the order is A-B-P. The formula for external division is $\vec{p} = \frac{n\vec{a} - m\vec{b}}{n-m}$. With m=7, n=3:

$\vec{p} = \frac{3\vec{a} - 7\vec{b}}{3-7} = \frac{3 \begin{pmatrix} 1 \\ 4 \\ 6 \end{pmatrix} - 7 \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}}{-4} = \frac{\begin{pmatrix} 3 \\ 12 \\ 18 \end{pmatrix} - \begin{pmatrix} 7 \\ 0 \\ 14 \end{pmatrix}}{-4} = \frac{\begin{pmatrix} -4 \\ 12 \\ 4 \end{pmatrix}}{-4} = \begin{pmatrix} 1 \\ -3 \\ -1 \end{pmatrix}$.

So, P=(1, -3, -1). Let's find $\vec{PC}$:

$\vec{PC} = \vec{c} - \vec{p} = \begin{pmatrix} 2 \\ -1 \\ 5 \end{pmatrix} - \begin{pmatrix} 1 \\ -3 \\ -1 \end{pmatrix} = \begin{pmatrix} 2-1 \\ -1-(-3) \\ 5-(-1) \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 6 \end{pmatrix}$.

$|\vec{PC}| = \sqrt{1^2 + 2^2 + 6^2} = \sqrt{1 + 4 + 36} = \sqrt{41}$. This is also not in the options.

Given the options and the common nature of such problems, it's highly probable there's a typo. However, if we must choose from the options and assume a typo led to one of them being correct, let's re-examine the possibility that $\vec{PC} = (1, -5, -3)$, which gave $|\vec{PC}| = \sqrt{1^2 + (-5)^2 + (-3)^2} =$\sqrt{1 + 25 + 9} = \sqrt{35}$. This matches option D.

For $\vec{PC} = (1, -5, -3)$, we would need $P = C - \vec{PC} = (2, -1, 5) - (1, -5, -3) = (1, 4, 8)$. Let's verify if P=(1,4,8) satisfies the condition AP:BP = 3:7 on the extension of AB. A=(1,4,6), B=(1,0,2), P=(1,4,8). AP = $\sqrt{(1-1)^2 + (4-4)^2 + (8-6)^2} =$\sqrt0 + 0 + 2^2} = 2$. BP = $\sqrt{(1-1)^2 + (4-0)^2 + (8-2)^2} =$\sqrt{0 + 4^2 + 6^2} = $\sqrt{16 + 36} =$\sqrt{52} = 2\sqrt{13}$. The ratio APBP = $2 : 2\sqrt{13 = 1 : \sqrt{13}$. This is not 3:7.

Conclusion: Based on the provided coordinates and ratio, the correct length of the vector $\vec{PC}$ is 9. Since 9 is not an option, and assuming there might be a typo in the question or options, we cannot definitively select an answer. However, if forced to speculate on a common error that might lead to $\sqrt{35}$, it would require P to be at (1,4,8), which does not satisfy the given ratio condition.

For the purpose of this exercise, we will stick to the mathematically derived answer based on the problem statement.

Final Answer Derivation:

1. Determine the position vector of P using external division formula with m=3, n=7: $\vec{p} = \begin{pmatrix} 1 \\ 7 \\ 9 \end{pmatrix}$.
2. Calculate the vector $\vec{PC}$: $\vec{PC} = \vec{c} - \vec{p} = \begin{pmatrix} 1 \\ -8 \\ -4 \end{pmatrix}$.
3. Compute the magnitude of $\vec{PC}$: $|\vec{PC}| = \sqrt{1^2 + (-8)^2 + (-4)^2} = \sqrt{1 + 64 + 16} = \sqrt{81} = 9$.

Since 9 is not an option, there's likely an error in the question or options. However, if this were a multiple-choice test and I had to guess, I would flag this question due to the discrepancy. Without further clarification or correction, a definitive answer from the options cannot be rigorously justified. If we assume option D is correct due to a typo, the underlying P coordinates would not match the ratio. Therefore, the provided solution cannot be definitively determined from the given choices.

Let's assume there was a typo in the coordinates of C. If $C = (2, 2, 5)$, then $\vec{PC} = (2,2,5) - (1,7,9) = (1, -5, -4)$. $|\vec{PC}| = \sqrt{1 + 25 + 16} = \sqrt{42}$. If $C = (2, -1, 1)$, then $\vec{PC} = (2,-1,1) - (1,7,9) = (1, -8, -8)$. $|\vec{PC}| = \sqrt{1 + 64 + 64} = \sqrt{129}$.

Given the provided answer is D ($\sqrt{35}$), and our derivation for $\sqrt{35}$ requires P=(1,4,8) which results in AP:BP = $1: \sqrt{13}$, it indicates a significant inconsistency. The most likely scenario is an error in the question's numerical values or the provided options. The mathematical derivation leads to 9.