A vector a makes equal acute angles with the | vedclass

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

Question

(B) Let the equal acute angles of vector $a$ with the coordinate axes be $\alpha$. Since the direction cosines are $l = \cos \alpha$,$m = \cos \alpha$

Vedclass · Accepted Answer

$\frac{11}{\sqrt{3}}$

Vedclass · Answer

(B) Let the equal acute angles of vector $a$ with the coordinate axes be $\alpha$. Since the direction cosines are $l = \cos \alpha$,$m = \cos \alpha$,and $n = \cos \alpha$,we have the relation $l^2 + m^2 + n^2 = 1$. Substituting the values,we get $3 \cos^2 \alpha = 1$,which implies $\cos \alpha = \frac{1}{\sqrt{3}}$. Thus,the direction ratios of vector $a$ are proportional to $(1, 1, 1)$,so we can take $a = \hat{i} + \hat{j} + \hat{k}$. The projection of vector $b$ on vector $a$ is given by the formula $\frac{b \cdot a}{|a|}$. Calculating the dot product: $b \cdot a = (5\hat{i} + 7\hat{j} - \hat{k}) \cdot (\hat{i} + \hat{j} + \hat{k}) = 5(1) + 7(1) - 1(1) = 5 + 7 - 1 = 11$. The magnitude of vector $a$ is $|a| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$. Therefore,the projection is $\frac{11}{\sqrt{3}}$.

$A$ vector $a$ makes equal acute angles with the coordinate axes. Then the projection of vector $b = 5\hat{i} + 7\hat{j} - \hat{k}$ on $a$ is

Similar Questions

In a triangle $ABC$ with usual notations,if $|\overline{BC}|=8, |\overline{CA}|=7, |\overline{AB}|=10$,then the projection of $\overline{AB}$ on $\overline{AC}$ is

If $a^2 + b^2 + c^2 = 1$,then the maximum possible value of $3a + 4b + 12c$ is equal to (where $a, b, c \in \mathbb{R}$)-

Let $\bar{a} = 2\hat{i} + \hat{j} + \hat{k}$,$\bar{b} = \hat{i} + 2\hat{j} - \hat{k}$,and vector $\bar{c}$ be coplanar with $\bar{a}$ and $\bar{b}$. If $\bar{c}$ is perpendicular to $\bar{a}$,then $\bar{c}$ is:

$\overline{a}, \overline{b}$ and $\overline{c}$ are three vectors such that $\overline{a}+\overline{b}+\overline{c}=\overline{0}$ and $|\overline{a}|=3, |\overline{b}|=5, |\overline{c}|=7$. The angle between $\overline{a}$ and $\overline{b}$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module