Let two non-collinear unit vectors hat{a} and | vedclass

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

Question

(A) The position vector is $\overline{OP} = \hat{a} \cos t + \hat{b} \sin t$.The square of the length is $|\overline{OP}|^2 = (\hat{a} \cos t + \hat{b

Vedclass · Accepted Answer

$\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}$ and $M=(1+\hat{a} \cdot \hat{b})^{1/2}$

Vedclass · Answer

(A) The position vector is $\overline{OP} = \hat{a} \cos t + \hat{b} \sin t$.The square of the length is $|\overline{OP}|^2 = (\hat{a} \cos t + \hat{b} \sin t) \cdot (\hat{a} \cos t + \hat{b} \sin t) = \cos^2 t + \sin^2 t + 2 \sin t \cos t (\hat{a} \cdot \hat{b}) = 1 + \sin(2t) (\hat{a} \cdot \hat{b})$.Since $\hat{a}$ and $\hat{b}$ form an acute angle,$\hat{a} \cdot \hat{b} > 0$. Thus,$|\overline{OP}|$ is maximum when $\sin(2t) = 1$,i.e.,$2t = \frac{\pi}{2} \Rightarrow t = \frac{\pi}{4}$.At $t = \frac{\pi}{4}$,$M = |\overline{OP}| = (1 + \hat{a} \cdot \hat{b})^{1/2}$.The vector $\overline{OP}$ at $t = \frac{\pi}{4}$ is $\frac{1}{\sqrt{2}}(\hat{a} + \hat{b})$.The unit vector $\hat{u}$ is $\frac{\overline{OP}}{|\overline{OP}|} = \frac{\frac{1}{\sqrt{2}}(\hat{a} + \hat{b})}{\frac{1}{\sqrt{2}}|\hat{a} + \hat{b}|} = \frac{\hat{a} + \hat{b}}{|\hat{a} + \hat{b}|}$.

Similar Questions

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

$|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = \dots$

If $\overline{a}, \overline{b}, \overline{c}$ are three vectors,$|\overline{a}|=2, |\overline{b}|=4, |\overline{c}|=1$,$|\overline{b} \times \overline{c}|=\sqrt{15}$ and $\overline{b}=2 \overline{c}+\lambda \overline{a}$,then the value of $\lambda$ is

The set of real values of $\lambda$ for which the vectors $\lambda \hat{i}-3 \hat{j}+5 \hat{k}$ and $2 \lambda \hat{i}-\lambda \hat{j}+\hat{k}$ are perpendicular to each other is

Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{u} \times \hat{v}| = \frac{\sqrt{3}}{2}$. If $\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} \times \hat{v})$, then $\lambda$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module