Find the values of for which the angle between the vectors a=2 ^2 i+4 j+k and b=7 i-2 j+ k

If is the angle between a and b, then = a b |a||b| If is obtuse then 0 ] & (2 ^2 i+4 j+k ) (7 i-2 j+ k)<0 & 14 ^2-8 + <0 i.e. 14 ^2-7 <0 & 7 (2 -1)<0 i.e. ( -1 2 )<0 i.e. 0< <1 2 Thus the angle between a and b is abtuse if 0< <1 2

Find the values of for which the angle between the vectors a=2 ^2…

Download App

Question

Find the values of $\lambda$ for which the angle between the vectors $\vec{a}=2 \lambda^2 \hat{i}+4 \lambda \hat{j}+\hat{k}$ and $\vec{b}=7 \hat{i}-2 \hat{j}+\lambda \hat{k}$

✓

Answer

If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then $\cos \theta=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$
If $\theta$ is obtuse then $\cos \theta<0$
$
\begin{aligned}
& \left.\therefore \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}<0 \text { i.e. } \vec{a} \cdot \vec{b}<0 \quad \quad \quad \quad \because|\vec{a}||\vec{b}|>0\right] \\
& \therefore\left(2 \lambda^2 \hat{i}+4 \lambda \hat{j}+\hat{k}\right) \cdot(7 \hat{i}-2 \hat{j}+\lambda \hat{k})<0 \\
& \therefore 14 \lambda^2-8 \lambda+\lambda<0 \text { i.e. } 14 \lambda^2-7 \lambda<0 \\
& \therefore 7 \lambda(2 \lambda-1)<0 \text { i.e. } \lambda\left(\lambda-\frac{1}{2}\right)<0 \text { i.e. } 0<\lambda<\frac{1}{2}
\end{aligned}
$
Thus the angle between $\bar{a}$ and $\bar{b}$ is abtuse if $0<\lambda<\frac{1}{2}$

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 "Solve the Following Question.(2 Marks)" from Vectors→Vectors — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Vectors — Maths STD 12 Science — Question

Answer

Need a full question paper?

Explore more

Similar questions