Linear Algebra Examples

${[\begin{array}{c} - 1 \\ 1 \\ 0 \end{array}], [\begin{array}{c} 1 \\ 1 \\ - 1 \end{array}]}$

Step 1

Two vectors are orthogonal if the dot product of them is $0$ .

Step 2

Evaluate the dot product of $[\begin{array}{c} - 1 \\ 1 \\ 0 \end{array}]$ and $[\begin{array}{c} 1 \\ 1 \\ - 1 \end{array}]$ .

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Step 2.1

The dot product of two vectors is the sum of the products of the their components.

$- 1 \cdot 1 + 1 \cdot 1 + 0 \cdot - 1$

Step 2.2

Simplify.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $- 1$ by $1$ .

$- 1 + 1 \cdot 1 + 0 \cdot - 1$

Step 2.2.1.2

Multiply $1$ by $1$ .

$- 1 + 1 + 0 \cdot - 1$

Step 2.2.1.3

Multiply $0$ by $- 1$ .

$- 1 + 1 + 0$

Step 2.2.2

Add $- 1$ and $1$ .

$0 + 0$

Step 2.2.3

Add $0$ and $0$ .

$0$

Step 3

The vectors are orthogonal since the dot product is $0$ .

Orthogonal

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