Using vector method, prove that the point is collinear: A(6, -7, … - Vidyadip

Question

Using vector method, prove that the point is collinear:
A(6, -7, -1), B(2, -3, 1) and C(4, -5, 0)

✓

Answer

Given the points A(6, -7, -1), B(2, -3, 1) and C(4, -5, 0). Then, $\overrightarrow{\text{AB}}=$ Position vector of B - Position vector of A$=2\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}-6\hat{\text{i}}+7\hat{\text{j}}+\hat{\text{k}}$
$=-4\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}}$
$=-2\big(2\hat{\text{i}}-2\hat{\text{j}}-\hat{\text{k}}\big)$$\overrightarrow{\text{BC}}=$ Position vector of C - Position vector of B
$=4\hat{\text{i}}-5\hat{\text{j}}-2\hat{\text{k}}+3\hat{\text{j}}-\hat{\text{k}}$
$=2\hat{\text{i}}-2\hat{\text{j}}-\hat{\text{k}}$
$\therefore\ \overrightarrow{\text{AB}}=-2\overrightarrow{\text{BC}}$
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 "5 Marks Questions" from VECTOR ALGEBRA→VECTOR ALGEBRA — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\text{A}=\begin{bmatrix}0&\text{c}&-\text{b}\\-\text{c}&0&\text{a}\\\text{b}&-\text{a}&0\end{bmatrix}$ and $\text{B}=\begin{bmatrix}\text{a}^2&\text{ab}&\text{ac}\\\text{ab}&\text{b}^2&\text{bc}\\\text{ac}&\text{bc}&\text{c}^2\end{bmatrix},$ show that $AB = BA = O_{3 \times 3}$

Evaluate the following intregals:
$\int\frac{\text{x}^2}{\text{x}^4-\text{x}^2-12}\ \text{dx}$

Using properties of definite integrals, evaluate:
$\int\limits^{\pi/4}_{0}\text{log (1 + tan x) dx}$.

If $\text{y}\sin(\text{x}^\text{x}),$ prove that $\frac{\text{dy}}{\text{dx}}=\cos(\text{x}^\text{x})\times\text{x}^\text{x}(1+\log\text{x})$

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).

Solve the following initial value problems:
$\text{x}(\text{x}^2+3\text{y}^2)\text{dx}+\text{y}(\text{y}^2+3\text{x}^2)\text{dy}=0,\text{y}(1)=1$

Find the condition that the curves $2x = y^2 $ and $2xy = k$ intersect orthogonally.

If $\text{x}^\text{m}.\text{y}^\text{n}=(\text{x}+\text{y})^{\text{m}+\text{n}},$ prove that:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}$

Find the area of the region $\big\{\text{(x, y)} : \text{x}^{2} +\text{y}^{2}\leq4, \ \text{x + y}\geq2,\big\}$ using the method of integration.

A manufacturer makes two types $A$ and $B$ of tea-cups. Three machines are needed for the manufacture and the time in minutes required for each cup on the machines is given below:

	Machines
	$I$	$II$	$III$
$A$	$12$	$18$	$6$
$B$	$6$	$0$	$9$

Each machine is available for a maximum of $6$ hours per day. If the profit on each cup $A$ is $75$ paise and that on each cup $B$ is $50$ paise, show that $15$ tea-cups of type $A$ and $30$ of type $B$ should be manufactured in a day to get the maximum profit.