# Linear Algebra Examples

Find the Angle Between the Vectors Using Dot Product
,
Step 1
Use the dot product formula to find the angle between two vectors.
Step 2
Find the dot product.
Step 2.1
The dot product of two vectors is the sum of the products of the their components.
Step 2.2
Simplify.
Step 2.2.1
Simplify each term.
Step 2.2.1.1
Multiply by .
Step 2.2.1.2
Multiply by .
Step 2.2.2
Step 3
Find the magnitude of .
Step 3.1
The norm is the square root of the sum of squares of each element in the vector.
Step 3.2
Simplify.
Step 3.2.1
One to any power is one.
Step 3.2.2
Raising to any positive power yields .
Step 3.2.3
Step 3.2.4
Any root of is .
Step 4
Find the magnitude of .
Step 4.1
The norm is the square root of the sum of squares of each element in the vector.
Step 4.2
Simplify.
Step 4.2.1
Raising to any positive power yields .
Step 4.2.2
One to any power is one.
Step 4.2.3
Step 4.2.4
Any root of is .
Step 5
Substitute the values into the formula.
Step 6
Simplify.
Step 6.1
Multiply by .
Step 6.2
Divide by .
Step 6.3
The exact value of is .