# Linear Algebra Examples

, ,

Step 1

Two vectors are orthogonal if the dot product of them is .

Step 2

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

Add and .

Step 2.2.3

Subtract from .

Step 3

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

Add and .

Step 3.2.3

Subtract from .

Step 4

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

Add and .

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

Add and .

Step 5

The vectors are orthogonal since the dot products are all .

Orthogonal