Using vector method, prove that the point is collinear: A(-3, -2,… - Vidyadip

Question

Using vector method, prove that the point is collinear:
A(-3, -2, -5), B(1, 2, 3) and C(3, 4, 7)

✓

Answer

Given the points A(-3, -2, -5), B(1, 2, 3) and C(3, 4, 7). Then,

$\overrightarrow{\text{AB}}=$ Position vector of B - Position vector of A

$=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}+3\hat{\text{i}}+2\hat{\text{j}}+5\hat{\text{k}}$

$=4\hat{\text{i}}+4\hat{\text{j}}+8\hat{\text{k}}$

$=2\big(2\hat{\text{i}}+2\hat{\text{j}}+4\hat{\text{k}}\big)$

$\overrightarrow{\text{BC}}=$ Position vector of C - Position vector of B

$=3\hat{\text{i}}+4\hat{\text{j}}+7\hat{\text{k}}-\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}}$

$=2\hat{\text{i}}+2\hat{\text{j}}+4\hat{\text{k}}$

$\therefore\ \overrightarrow{\text{AB}}=2\overrightarrow{\text{BC}}$

So, $\overrightarrow{\text{AB}}$ and $\overrightarrow{\text{BC}}$ are parallel vectors. But B is a point common to them.

Hence, the given points A, B, and C are collinear.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Algebra of Vectors→Algebra of Vectors — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Determine the maximum value of $\text{Z}=11\text{x}+7\text{y}$ subject to the constraints:
$2\text{x}+\text{y}\leq6,\text{x}\leq2,\text{x}\geq0,\text{y}\geq0. $

Differentiate the following functions with respect to x:
$\log\Big(\frac{\sin\text{x}}{1+\cos\text{x}}\Big)$

If $\vec{\text{a}}=\vec{\text{i}}+\vec{\text{j}}+2\vec{\text{k}}$ and $\vec{\text{b}}=2\vec{\text{i}}+\vec{\text{j}}-2\vec{\text{k}},$ find the unit vector in the direction of:

$6\vec{\text{b}}$
$2\vec{\text{a}}-\vec{\text{b}}$

Maximise and Minimise Z = 3x – 4y subject to:
$\text{x}-2\text{y}\leq0$
$-3\text{x}+\text{y}\leq4$
$\text{x}-\text{y}\leq6$
$\text{x},\text{y}\geq0$

Evaluate the following integrals:
$\int\text{x}\Big(\frac{\sec2\text{x}-1}{\sec2\text{x}+1}\Big)\text{dx}$

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}6&-3&2\\2&-1&2\\-10&5&2 \end{vmatrix}$

Without using the concept of inverse of a matrix, find the matrix $\begin{bmatrix}\text{x}&\text{y}\\\text{z}&\text{u}\end{bmatrix}$ such that $\begin{bmatrix}5&-7\\-2&3\end{bmatrix}\begin{bmatrix}\text{x}&\text{y}\\\text{z}&\text{u}\end{bmatrix}=\begin{bmatrix}-16&-6\\7&2\end{bmatrix}$

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}+\frac{1+\text{y}^2}{\text{y}}=0$

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
A = The number of heads is odd,
B = The number of tails is odd.

Evaluate the following integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\sqrt{\sin\phi}\cos^5\phi\text{ d}\phi$