Trigonometry Examples

Find the Angle Between the Vectors (1,-2) , (-6,-3)
,
Step 1
Use the dot product formula to find the angle between two vectors.
Step 2
Find the dot product.
Tap for more steps...
Step 2.1
The dot product of two vectors is the sum of the products of the their components.
Step 2.2
Simplify.
Tap for more steps...
Step 2.2.1
Simplify each term.
Tap for more steps...
Step 2.2.1.1
Multiply by .
Step 2.2.1.2
Multiply by .
Step 2.2.2
Add and .
Step 3
Find the magnitude of .
Tap for more steps...
Step 3.1
The norm is the square root of the sum of squares of each element in the vector.
Step 3.2
Simplify.
Tap for more steps...
Step 3.2.1
One to any power is one.
Step 3.2.2
Raise to the power of .
Step 3.2.3
Add and .
Step 4
Find the magnitude of .
Tap for more steps...
Step 4.1
The norm is the square root of the sum of squares of each element in the vector.
Step 4.2
Simplify.
Tap for more steps...
Step 4.2.1
Raise to the power of .
Step 4.2.2
Raise to the power of .
Step 4.2.3
Add and .
Step 4.2.4
Rewrite as .
Tap for more steps...
Step 4.2.4.1
Factor out of .
Step 4.2.4.2
Rewrite as .
Step 4.2.5
Pull terms out from under the radical.
Step 5
Substitute the values into the formula.
Step 6
Simplify.
Tap for more steps...
Step 6.1
Cancel the common factor of and .
Tap for more steps...
Step 6.1.1
Factor out of .
Step 6.1.2
Cancel the common factors.
Tap for more steps...
Step 6.1.2.1
Factor out of .
Step 6.1.2.2
Cancel the common factor.
Step 6.1.2.3
Rewrite the expression.
Step 6.2
Simplify the denominator.
Tap for more steps...
Step 6.2.1
Raise to the power of .
Step 6.2.2
Raise to the power of .
Step 6.2.3
Use the power rule to combine exponents.
Step 6.2.4
Add and .
Step 6.3
Rewrite as .
Tap for more steps...
Step 6.3.1
Use to rewrite as .
Step 6.3.2
Apply the power rule and multiply exponents, .
Step 6.3.3
Combine and .
Step 6.3.4
Cancel the common factor of .
Tap for more steps...
Step 6.3.4.1
Cancel the common factor.
Step 6.3.4.2
Rewrite the expression.
Step 6.3.5
Evaluate the exponent.
Step 6.4
Divide by .
Step 6.5
The exact value of is .