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 + kThe 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€“ kThe dot product is defined as

\(\vec{a}. \vec{b}\)= (5i – j + k)(i + j – k)

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

\(\theta = cos^{-1}\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}\)\(\theta = cos^{-1}\frac{3}{(5.19)(1.73)}\)

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

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

\(\theta =70.48^{\circ}\)

To learn more formulas on different concepts, visit BYJU’S – The Learning App and download the app to learn with ease.