Download App

VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA5 Marks
Question
Show that the points whose position vectors are$\vec{\text{a}}=4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}, \vec{\text{b}}=2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}},\vec{\text{c}}=\hat{\text{i}}-\hat{\text{j}}$ from a right triangle.
✓

Answer

Given
$\vec{\text{a}}=4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}$
$\vec{\text{b}}=2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$
$\vec{\text{c}}=\hat{\text{i}}-\hat{\text{j}}$
$\overrightarrow{\text{AB}}$ = position vector of B - position vector of A
$=\big(2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}\big)-\big(4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}\big)$
$=2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}-4\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}$
$\overrightarrow{\text{AB}}=-2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}$
$\overrightarrow{\text{BC}}$ = position vector of C - position vector of B
$=\big(\hat{\text{i}}-\hat{\text{j}}\big)-\big(2\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}\big)$
$=\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{i}}+4\hat{\text{j}}-5\hat{\text{k}}$
$=-\hat{\text{i}}+3\hat{\text{j}}-5\hat{\text{k}}$
$\overrightarrow{\text{CA}}$ = position vector of A - position vector of C
$=\big(4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}\big)-\big(\hat{\text{i}}-\hat{\text{j}}\big)$
$=4\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}}-\hat{\text{i}}+\hat{\text{j}}$
$=3\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$

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

Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\text{e}^{\theta}\Big(\theta+\frac{1}{\theta}\Big)\text{ and y}=\text{e}^{-\theta}\Big(\theta-\frac{1}{\theta}\Big)$
→
Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-5}{4}=\frac{\text{y}-7}{4}=\frac{\text{z}+3}{-5}$ and $\frac{\text{x}-8}{7}=\frac{\text{y}-4}{1}=\frac{3-5}{3}$
→
Show that the points A, B, C with position vectors $2\hat{\text{i}} - \hat{\text{j}} + \hat{\text{k}}, \hat{\text{i}} - 3\hat{\text{j}} - 5\hat{\text{k}} \text{ and } 3\hat{\text{i}} - 4\hat{\text{j}} - 4\hat{\text{k}}$ respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle.
→
Draw the rough sketch of $\frac{\text{x}^{2}}{4}+\frac{\text{y}^{2}}{9}=1$ and evaluate the area of the region under the area the curve and the line x-axis.
→
Find the points on the curve $y^2 = 2x^3 $ at which the slope of the tangent is $3.$
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\frac{(\text{x}^2-1)^3(2\text{x}-1)}{\sqrt{(\text{x}-3)(4\text{x}-1)}}$
→
If $\begin{vmatrix}\text{p}&\text{q}&\text{c}\\\text{a}&\text{q}&\text{c}\\\text{a}&\text{b}&\text{r} \end{vmatrix}=0,$ find the value of $\frac{\text{p}}{\text{p}-\text{a}}+\frac{\text{q}}{\text{q}-\text{b}}+\frac{\text{r}}{\text{r}-\text{c}},$ $\text{p}\neq\text{a},\text{q}\neq\text{b},\text{r}\neq\text{c}.$
→
A box contains 100 tickets, each bearing one of the numbers from 1 to 100. If 5 tickets are drawn successively with replacement from the box, find the probability that all the tickets bear numbers divisible by 10.
→
Evaluate the following integrals as limit of sum:
$\int\limits^{\frac{\pi}{2}}_{0}\sin\text{x dx}$
→
The scalar product of the vector $\hat{\text{l}}+\hat{\text{j}}+\hat{\text{k}}$ with a unit a vector along the sum of vectors $2\hat{\text{l}}+4\hat{\text{j}}-5\hat{\text{k}}\ \text{and}\ \lambda\hat{\text{l}}+2\hat{\text{j}}+3\hat{\text{k}}$ is equal to one. Find the value of $\lambda.$
→