Using vector method, prove that the point is collinear: A(2, -1, … - Vidyadip

Question

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

✓

Answer

Given the points A(2, -1, 3), B(4, 3, 1) and C(3, 1, 2). Then, $\overrightarrow{\text{AB}}=$ Position vector of B - Position vector of A$=4\hat{\text{i}}+3\hat{\text{j}}+\hat{\text{k}}-2\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}$
$=2\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}$
$=-2\big(-\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}\big)$$\overrightarrow{\text{BC}}=$ Position vector of C - Position vector of B
$=3\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}-4\hat{\text{i}}-3\hat{\text{j}}-\hat{\text{k}}$
$=-\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$
$\therefore\ \overrightarrow{\text{AB}}=-2\overrightarrow{\text{BC}}$
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 "Solve the Following Question.(3 Marks)" from Algebra of Vectors→Algebra of Vectors — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If $\big|\vec{\text{a}}\times\vec{\text{b}}\big|^2+\big|\vec{\text{a}}.\vec{\text{b}}\big|^2=400$ and $|\vec{\text{a}}|=5,$ then write the value of $\big|\vec{\text{b}}\big|.$

Classify the following functions as injection, surjection or bijection: $f : R \rightarrow R,$ defined by $f(x) = 1 + x^2$

Show that $\text{y}=\text{ax}^3+\text{bx}^2+\text{c}$ is a solution of the differential equation $\frac{\text{d}^3\text{y}}{\text{dx}^3}=6\text{a}$

Show that the line through the points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (-1, -2, 1) and, (1, 2, 5).

Evaluate the following integrals:$\int^\limits6_{-6}\big|\text{x}+2\big|\text{dx}$

Find the area of the region bounded by the parabola:

$y^2=16 x$ and its latus rectum.

If a machine is correctly set up it produces $90\%$ acceptable items. If it is incorrectly set up it produces only $40%$ acceptable item. Past experience shows that $80\%$ of the setups are correctly done. If after a certain set up, the machine produces $2$ acceptable items, find the probability that the machine is correctly set up.

The volume of the spherical ball is increasing at the rate of $4 \pi \mathrm{cc} / \mathrm{sec}$. Find the rate at which the radius and the surface area are changing when the volume is $288 \pi \mathrm{cc}$.

ABCDE is a pentagon, prove that,$\overrightarrow{\text{AB}}+\overrightarrow{\text{AE}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{DC}}+\overrightarrow{\text{ED}}+\overrightarrow{\text{AC}}=3\ \overrightarrow{\text{AC}}$

Determine which of the following binary operations are associative and which are commutative:
* on Q defined by $\text{a}\ ^*\ \text{b}=\frac{\text{a}+\text{b}}{2}$ for all $\text{a, b}\in\text{Q}$