# Linear Algebra Examples

Determine if the Vectors are Orthogonal
, ,
Step 1
Two vectors are orthogonal if the dot product of them is .
Step 2
Evaluate the dot product of and .
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 2.2.1.3
Multiply by .
Step 2.2.2
Step 2.2.3
Subtract from .
Step 3
Evaluate the dot product of and .
Step 3.1
The dot product of two vectors is the sum of the products of the their components.
Step 3.2
Simplify.
Step 3.2.1
Simplify each term.
Step 3.2.1.1
Multiply by .
Step 3.2.1.2
Multiply .
Step 3.2.1.2.1
Multiply by .
Step 3.2.1.2.2
Multiply by .
Step 3.2.1.3
Multiply by .
Step 3.2.2
Step 3.2.3
Subtract from .
Step 4
Evaluate the dot product of and .
Step 4.1
The dot product of two vectors is the sum of the products of the their components.
Step 4.2
Simplify.
Step 4.2.1
Simplify each term.
Step 4.2.1.1
Multiply by .
Step 4.2.1.2
Multiply .
Step 4.2.1.2.1
Raise to the power of .
Step 4.2.1.2.2
Raise to the power of .
Step 4.2.1.2.3
Use the power rule to combine exponents.
Step 4.2.1.2.4
Step 4.2.1.3
Rewrite as .
Step 4.2.1.3.1
Use to rewrite as .
Step 4.2.1.3.2
Apply the power rule and multiply exponents, .
Step 4.2.1.3.3
Combine and .
Step 4.2.1.3.4
Cancel the common factor of .
Step 4.2.1.3.4.1
Cancel the common factor.
Step 4.2.1.3.4.2
Rewrite the expression.
Step 4.2.1.3.5
Evaluate the exponent.
Step 4.2.1.4
Multiply by .
Step 4.2.1.5
Multiply by .
Step 4.2.2
Subtract from .
Step 4.2.3