Question

Given two vectors: A = 2i + 3j and B = -3i + 2j. Find the angle between the two vectors.​

Answer 1

Given,

Two vectors:

A = 2i + 3j,

B = -3i + 2j.

To find,

The angle between vectors A and B.

Solution,

The angle between any two vectors can be determined by using the formula for the scalar or dot product of the given vectors.

The scalar or dot product of two vectors is given as,

$\vec a. \vec b =|\vec a|.|\vec b|\cos\theta$

Where,

$|\vec a| \hspace{3} and \hspace{3} |\vec b| =$ are the magnitudes of vectors a and b, and,

θ = the angle between them.

Now, on rearranging this equation, we get,

$\cos\theta=\frac{\vec a. \vec b}{|\vec a|.|\vec b|}$

which can be used to find the angle between the two vectors.

For the given vectors A and B,

$\vec A. \vec B =(2i + 3j).(-3i + 2j)$

⇒ $\vec A. \vec B =2(-3)+3(2)$

⇒ $\vec A. \vec B = -6+6=0$

Also,

$|\vec A| =\sqrt{2^2+3^2} = \sqrt{13}$, and,

$|\vec B| =\sqrt{(-3)^2+2^2} = \sqrt{13}$

Now, since $\vec A. \vec B =0$

⇒ $\cos\theta=\frac{\vec A. \vec B}{|\vec A|.|\vec B|}=0$

⇒ $\theta =\cos^{-1}(0)$

⇒ θ = 90°.

Therefore, the angle between the given vectors A and B is 90°.