Download App

VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA5 Marks
Question
Let $\vec{\text{a}}=\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}},\vec{\text{b}}=3\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}$ and $\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}.$Find a vector $\vec{\text{d}}$ which is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{d}}$ and $\vec{\text{c}}.\vec{\text{d}}=15.$
✓

Answer

Given
$\vec{\text{a}}=\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}}$
$\vec{\text{b}}=3\hat{\text{i}}-2\hat{\text{j}}+7\hat{\text{k}}$
$\vec{\text{c}}=2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}$
Since d is perpendicular to both a and b, it is parallel to $\vec{\text{a}}\times\vec{\text{b}}.$
Suppose $\text{d}=\lambda\big(\vec{\text{a}}\times\vec{\text{b}}\big)$ for some scalar $\lambda.$
$\text{d}=\lambda\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}}\\1&4&2\\3&-2&7 \end{vmatrix}$
$=\lambda\big[(28+4)\hat{\text{i}}-(7-6)\hat{\text{j}}+(-2-12)\hat{\text{k}}\big]$
$\vec{\text{c}}.\vec{\text{d}}=15$ (Given)
$\Rightarrow\big(2\hat{\text{i}}-\hat{\text{j}}+4\hat{\text{k}}\big).\lambda\big(32\hat{\text{i}}-\hat{\text{j}}-14\hat{\text{k}}\big)=15$
$\Rightarrow\lambda(64+1-56)=15$
$\Rightarrow\lambda=\frac{5}{3}$
$\therefore\vec{\text{d}}=\frac{5}{3}\big(32\hat{\text{i}}-\hat{\text{j}}-14\hat{\text{k}}\big)$
$\Rightarrow\vec{\text{d}}=\frac{1}{3}\big(160\hat{\text{i}}-5\hat{\text{j}}-70\hat{\text{k}}\big)$
Disclaimer: The question should contain "Which is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{b}}$ " instead of "Which is perpendicular to both $\vec{\text{a}}$ and $\vec{\text{d}}$ "

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

$\text{Let}\ \vec{\text{a}}=4\hat{\text{i}}+5\hat{\text{j}}-\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}-4\hat{\text{j}}+5\hat{\text{k}}$ and $\vec{\text{c}}=3\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ Find a vector $\vec{\text{d}}$ which is perpendicular to both $\vec{\text{c}}\ \text{and }\vec{\text{b}}\ \text{and}\ \vec{\text{d}}\cdot\vec{\text{a}}=21.$
→
Evaluate the following integrals:
$\int\sec^4\text{x}\tan\text{x}\text{ dx}$
→
Evaluate the following integrals:
$\int\frac{1}{\sin\text{x}(3+2\cos\text{x})}\ \text{dx}$
→
Find the condition that the curves $2x = y^2 $ and $2xy = k$ intersect orthogonally.
→
Find the points on the curve $y = x^3 - 2x^2 - 2x$ at which the tangent lines are parallel to the line $y = 2x - 3.$
→
In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater then 4. Find the expected value of the amount he wins or loses.
→
Determine the maximum distance that the man can travel.
→
Find the reflection of the point (1, 2, -1) in the plane 3x - 5y + 4z = 5.
→
Differentiate $\frac{\text{x}}{\sin\text{x}}$ w.r.t. $\sin\text{x}.$
→
Find the equation of the plane through the point $2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}$ and passing throught the line of intersection of the planes $\vec{\text{r}}\cdot(\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}})=0$ and $\vec{\text{r}}\cdot(\hat{\text{j}}+2\hat{\text{k}})=0.$
→