Let vec{alpha}=4 hat{i}+3 hat{j}+5 hat{k} and | vedclass

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

Question

(C) Let $\vec{\beta}_1 = \lambda \vec{\alpha}$.Since $\vec{\beta} = \vec{\beta}_1 + \vec{\beta}_2$,we have $\vec{\beta}_2 = \vec{\beta} - \vec{\beta}_

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(C) Let $\vec{\beta}_1 = \lambda \vec{\alpha}$.Since $\vec{\beta} = \vec{\beta}_1 + \vec{\beta}_2$,we have $\vec{\beta}_2 = \vec{\beta} - \vec{\beta}_1 = \vec{\beta} - \lambda \vec{\alpha}$.Substituting the vectors,$\vec{\beta}_2 = (\hat{i} + 2\hat{j} - 4\hat{k}) - \lambda(4\hat{i} + 3\hat{j} + 5\hat{k}) = (1 - 4\lambda)\hat{i} + (2 - 3\lambda)\hat{j} - (4 + 5\lambda)\hat{k}$.Since $\vec{\beta}_2 \perp \vec{\alpha}$,their dot product is zero: $\vec{\beta}_2 \cdot \vec{\alpha} = 0$.$4(1 - 4\lambda) + 3(2 - 3\lambda) + 5(-4 - 5\lambda) = 0$.$4 - 16\lambda + 6 - 9\lambda - 20 - 25\lambda = 0$.$-50\lambda - 10 = 0 \Rightarrow \lambda = -\frac{1}{5}$.Now,$\vec{\beta}_2 = (1 - 4(-\frac{1}{5}))\hat{i} + (2 - 3(-\frac{1}{5}))\hat{j} - (4 + 5(-\frac{1}{5}))\hat{k} = \frac{9}{5}\hat{i} + \frac{13}{5}\hat{j} - 3\hat{k}$.Thus,$5\vec{\beta}_2 = 9\hat{i} + 13\hat{j} - 15\hat{k}$.Finally,$5\vec{\beta}_2 \cdot (\hat{i} + \hat{j} + \hat{k}) = 9(1) + 13(1) - 15(1) = 9 + 13 - 15 = 7$.

Similar Questions

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\hat{c}$ is :

If $P(3, 4, 5)$,$Q(4, 6, 3)$,$R(-1, 2, 4)$,and $S(1, 0, 5)$,then the projection of the vector $\vec{RS}$ on the vector $\vec{PQ}$ is:

If $a\hat{i} + 6\hat{j} - \hat{k}$ and $7\hat{i} - 3\hat{j} + 17\hat{k}$ are perpendicular vectors,then what is the value of $a$?

If $\vec{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}, \vec{b}=-\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=3 \hat{i}+\hat{j}$ are such that $\vec{a}+\lambda \vec{b}$ is perpendicular to $\vec{c},$ then find the value of $\lambda$.

The vector $2i + j - k$ is perpendicular to $i - 4j + \lambda k,$ if $\lambda = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module