# Linear Algebra Examples

Find the Angle Between the Vectors Using Cross Product
,
Step 1
Use the cross product formula to find the angle between two vectors.
Step 2
Find the cross product.
Step 2.1
The cross product of two vectors and can be written as a determinant with the standard unit vectors from and the elements of the given vectors.
Step 2.2
Set up the determinant with the given values.
Step 2.3
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in row by its cofactor and add.
Step 2.3.1
Consider the corresponding sign chart.
Step 2.3.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 2.3.3
The minor for is the determinant with row and column deleted.
Step 2.3.4
Multiply element by its cofactor.
Step 2.3.5
The minor for is the determinant with row and column deleted.
Step 2.3.6
Multiply element by its cofactor.
Step 2.3.7
The minor for is the determinant with row and column deleted.
Step 2.3.8
Multiply element by its cofactor.
Step 2.3.9
Step 2.4
Evaluate .
Step 2.4.1
The determinant of a matrix can be found using the formula .
Step 2.4.2
Simplify the determinant.
Step 2.4.2.1
Simplify each term.
Step 2.4.2.1.1
Multiply by .
Step 2.4.2.1.2
Multiply by .
Step 2.4.2.2
Subtract from .
Step 2.5
Evaluate .
Step 2.5.1
The determinant of a matrix can be found using the formula .
Step 2.5.2
Simplify the determinant.
Step 2.5.2.1
Simplify each term.
Step 2.5.2.1.1
Multiply by .
Step 2.5.2.1.2
Multiply by .
Step 2.5.2.2
Step 2.6
Evaluate .
Step 2.6.1
The determinant of a matrix can be found using the formula .
Step 2.6.2
Simplify the determinant.
Step 2.6.2.1
Simplify each term.
Step 2.6.2.1.1
Multiply by .
Step 2.6.2.1.2
Multiply by .
Step 2.6.2.2
Step 2.7
Multiply by .
Step 2.8
Step 3
Find the magnitude of the cross product.
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
Raise to the power of .
Step 3.2.2
Raise to the power of .
Step 3.2.3
Raise to the power of .
Step 3.2.4
Step 3.2.5
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
One to any power is one.
Step 4.2.2
Raise to the power of .
Step 4.2.3
Raise to the power of .
Step 4.2.4
Step 4.2.5
Step 5
Find the magnitude of .
Step 5.1
The norm is the square root of the sum of squares of each element in the vector.
Step 5.2
Simplify.
Step 5.2.1
Raising to any positive power yields .
Step 5.2.2
Raise to the power of .
Step 5.2.3
One to any power is one.
Step 5.2.4
Step 5.2.5
Step 6
Substitute the values into the formula.
Step 7
Simplify.
Step 7.1
Simplify the denominator.
Step 7.1.1
Combine using the product rule for radicals.
Step 7.1.2
Multiply by .
Step 7.2
Simplify the denominator.
Step 7.2.1
Rewrite as .
Step 7.2.1.1
Factor out of .
Step 7.2.1.2
Rewrite as .
Step 7.2.2
Pull terms out from under the radical.
Step 7.3
Multiply by .
Step 7.4
Combine and simplify the denominator.
Step 7.4.1
Multiply by .
Step 7.4.2
Move .
Step 7.4.3
Raise to the power of .
Step 7.4.4
Raise to the power of .
Step 7.4.5
Use the power rule to combine exponents.
Step 7.4.6
Step 7.4.7
Rewrite as .
Step 7.4.7.1
Use to rewrite as .
Step 7.4.7.2
Apply the power rule and multiply exponents, .
Step 7.4.7.3
Combine and .
Step 7.4.7.4
Cancel the common factor of .
Step 7.4.7.4.1
Cancel the common factor.
Step 7.4.7.4.2
Rewrite the expression.
Step 7.4.7.5
Evaluate the exponent.
Step 7.5
Simplify the numerator.
Step 7.5.1
Combine using the product rule for radicals.
Step 7.5.2
Multiply by .
Step 7.6
Multiply by .
Step 7.7
Evaluate .