# Angle between Two Vectors Formula

If the two vectors are assumed as $$\vec{a}$$ and $$\vec{b}$$ then the dot created is articulated as $$\vec{a}. \vec{b}$$. Letâ€™s suppose these two vectors are separated by angleÂ Î¸.Â To know what’s the angleÂ measurement we solve with the below formula

we know that the dot product of two product is given as

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

Thus, the angle between two vectors formula is given by

$$\theta = cos^{-1}\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$$

whereÂ Î¸ is the angle between $$\vec{a}$$ and $$\vec{b}$$

## Angle Between Two Vectors Examples

Letâ€™s see some samples on the angle between two vectors:

Example 1:

Â Compute the angle between two vectors 3i + 4j – k and 2i – j + k.

solution:

Let

$$\vec{a}$$ = 3i + 4j – k and

$$\vec{b}$$ = 2i – j + k

The dot product is defined as

$$\vec{a}. \vec{b}$$ = (3i + 4j – k).(2i – j + k)

= (3)(2) + (4)(-1) + (-1)(1)

= 6-4-1

= 1

Thus,Â $$\vec{a}. \vec{b}$$Â  = 1

The Magnitude of vectors is given by

$$|\vec{a}| =\sqrt{(3^{2}+4^{2}+(-1)^{2})} =\sqrt{26}= 5.09$$

$$|\vec{b}| =\sqrt{(2^{2}+(-1)^{2}+1^{2})} =\sqrt{6}= 2.45$$

The angle between the two vectors is

$$\theta = cos^{-1}\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$$

$$\theta = cos^{-1}\frac{1}{(5.09)(2.45)}$$

$$\theta = cos^{-1}\frac{1}{12.47}$$

$$\theta = cos^{-1}(0.0802)$$

$$\theta = 85.39^{\circ}$$

Example 2:Â

Find the angle between two vectors 5i – j + k and i + j – k.

Solution:

Let

$$\vec{a}$$ = 5i – j + k and

$$\vec{b}$$ = i + j â€“ k

The dot product is defined as
$$\vec{a}. \vec{b}$$= (5i – j + k)(i + j – k)

$$\vec{a}. \vec{b}$$= (5)(1) + (-1)(1) + (1)(-1)

$$\vec{a}. \vec{b}$$= 5-1-1

$$\vec{a}. \vec{b}$$= 3

The Magnitude of vectors is given by

$$|\vec{a}| =\sqrt{(5^{2}+(-1)^{2}+1^{2})} =\sqrt{27}= 5.19$$

$$|\vec{b}| =\sqrt{(1^{2}+1^{2}+(-1)^{2})} =\sqrt{3}= 1.73$$

The angle between the two vectors is

$$\theta = cos^{-1}\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$$

$$\theta = cos^{-1}\frac{3}{(5.19)(1.73)}$$

$$\theta = cos^{-1}\frac{3}{8.97}$$

$$\theta = cos^{-1}(0.334)$$

$$\theta =70.48^{\circ}$$