If the two vectors are assumed as a⃗ and b⃗ then the dot created is articulated as a.b. Let’s suppose these two vectors are separated by angle θ. To know what’s the angle measurement we solve with the below formula

The angle between two vectors formula is given by

where θ is the angle between a⃗ a→ and b⃗ b→.

**Examples**

**Let’s see some samples on angle between two vectors:**

**Problem 1: **Compute the angle between two vectors 3i + 4j – k and 2i – j + k.

**Answer:**

a⃗ = 3i + 4j – k and

b⃗ = 2i – j + k

The dot product is articulated as

a.b = (3i + 4j – k)(2i – j + k)

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

= 6-4-1

= 1

The Magnitude of vectors is given by

The angle between two vectors is

**Problem 2: **Find the angle between two vectors 5i – j + k and i + j – k.

**Answer:**

Known:

a⃗ = 5i – j + k and

b⃗ = i + j – k

The dot product is articulated as

a.b = (5i – j + k)(i + j – k)

a.b = (5)(1) + (-1)(1) + (1)(-1)

a.b = 5-1-1

a.b = 3

The Magnitude of vectors is given by

The angle between two vectors is