Let vec{a} = 3hat{i} + 2hat{j} + 2hat{k} and | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) First,calculate $\vec{a} + \vec{b} = (3+1)\hat{i} + (2+2)\hat{j} + (2-2)\hat{k} = 4\hat{i} + 4\hat{j}$.Next,calculate $\vec{a} - \vec{b} = (3-1)\h

Vedclass · Accepted Answer

$4(2\hat{i} - 2\hat{j} - \hat{k})$

Vedclass · Answer

(A) First,calculate $\vec{a} + \vec{b} = (3+1)\hat{i} + (2+2)\hat{j} + (2-2)\hat{k} = 4\hat{i} + 4\hat{j}$.Next,calculate $\vec{a} - \vec{b} = (3-1)\hat{i} + (2-2)\hat{j} + (2-(-2))\hat{k} = 2\hat{i} + 4\hat{k}$.$A$ vector perpendicular to both $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ is given by their cross product:$\vec{v} = (\vec{a} + \vec{b}) 	imes (\vec{a} - \vec{b}) = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 4 & 4 & 0 \ 2 & 0 & 4 \end{vmatrix} = \hat{i}(16 - 0) - \hat{j}(16 - 0) + \hat{k}(0 - 8) = 16\hat{i} - 16\hat{j} - 8\hat{k}$.Simplify the vector: $8(2\hat{i} - 2\hat{j} - \hat{k})$.The magnitude of this vector is $8 \sqrt{2^2 + (-2)^2 + (-1)^2} = 8 \sqrt{4+4+1} = 8 	imes 3 = 24$.We need a vector with magnitude $12$,which is half of the magnitude of $\vec{v}$.Thus,the required vector is $\pm \frac{12}{24} 	imes 8(2\hat{i} - 2\hat{j} - \hat{k}) = \pm 4(2\hat{i} - 2\hat{j} - \hat{k})$.Comparing with the options,$4(2\hat{i} - 2\hat{j} - \hat{k})$ is the correct choice.

Let $\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$ be two vectors. If a vector perpendicular to both the vectors $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ has the magnitude $12$,then one such vector is

Similar Questions

$a, b, c, d$ are coplanar vectors, then $(a \times b) \times (c \times d)$ is equal to

The direction cosines of a line which is perpendicular to lines whose direction ratios are $3, -2, 4$ and $1, 3, -2$ are

$A$ unit vector perpendicular to the vectors $\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}$ and $\vec{b} = 3 \hat{j} + 2 \hat{k}$ is

The points $A(a), B(b), C(c)$ will be collinear if

If the area of the triangle with vertices $\hat{i}+y \hat{j}$,$\hat{i}+2 \hat{k}$,and $3 \hat{j}+\hat{k}$ is $\sqrt{6}$ sq. units,then the values of $y$ are

Vedclass Test Series

Exam Paper Generator

Online Exam Module