Question

if a bar = 5i^-j^-3k^ and b bar =i^ +3j^-5k^ find the angle between a and b​

Answer 1

Answer:

Explanation: A*B will give us a new vector whose direction will be perpendicular to A and B. Thus the angle between A+B and A*B will be 90 degrees.

Answer 2

Given :

$\red{\bigstar}\:\sf{\vec{a}=5\hat{i}- \hat{j} -3\hat{k} }$

$\red{\bigstar}\:\sf{\vec{b}=\hat{i}+3\hat{j} - 5\hat{k}}$

To find :

$\red{\bigstar}\:\sf{angle\:between\:vector\:\vec{a} \;and\;\vec{b}}$

Formula required :

• Dot product of two vectors A and B is given by

$\boxed{\sf{\vec{A}\;.\;\vec{B}=\mid \vec{A}\mid . \mid \vec{B}\mid . cos\theta}}$

• Magnitude of a vector P with orthogonal notations

$\sf{Let,\;\vec{P}=a\hat{i} +b\hat{j}+c\hat{k}}$

$\boxed{\sf{then, \; \mid \vec{P} \mid = \sqrt{a^2 + b^2 + c^2}}}$

• dot product of two vectors with orthogonal notations

$\sf{Let, \; \vec{M} = a\hat{i} + b\hat{j} + c\hat{k} \;and\; \vec{N}=d\hat{i} + e\hat{j} + f\hat{k} }$

$\boxed{\sf{then,\; \vec{M} \;.\;\vec{N}=ad + be + cf }}$

Solution :

Calculating product of vector a with vector b

$\implies\sf{\vec{a}.\vec{b}=(5\hat{i} - \hat{j} - 3\hat{k}).(\hat{i}+3\hat{j}-5\hat{k})}$

$\implies\sf{\vec{a} . \vec{b}= (5\times 1) + (-1\times 3) + (-3\times -5)}$

$\implies\sf{\vec{a} . \vec{b}= 5 - 3 + 15}$

$\red{\implies\sf{\vec{a}.\vec{b}=17}}$

Calculating magnitude of vector a

$\implies\sf{\mid \vec{a}\mid =\sqrt{(5)^2+ (-1)^2+(-3)^2}}$

$\red{\implies\sf{\mid\vec{a}\mid=\sqrt{35}}}$

Calculating magnitude of vector b

$\implies\sf{\mid\vec{b}\mid=\sqrt{(1)^2+(3)^2+(-5)^2}}$

$\red{\implies\sf{\mid\vec{b}\mid=\sqrt{35}}}$

putting calculated values in formula for dot product of vectors

Let angle between vector a and b be θ

$\implies\sf{\vec{a}\;.\;\vec{b}=\mid \vec{a} \mid . \mid \vec{b} \mid . cos\theta}$

$\implies\sf{17=\sqrt{35}\; . \sqrt{35} \;.\; cos \theta}$

$\implies\sf{17=35 \;\; cos\theta}$

$\implies\sf{cos\theta=\dfrac{17}{35}}$

$\boxed{\boxed{\red{\implies\sf{\theta=cos^{-1} \left( \dfrac{17}{35}\right)^{\circ}}}}}\huge{\dagger}$