Linear Algebra Examples

⎧ ⎪ ⎨ ⎪ ⎩ ⎡ ⎢ ⎣ \begin{matrix} - 1 10 \end{matrix} ⎤ ⎥ ⎦, ⎡ ⎢ ⎣ \begin{matrix} 11 - 1 \end{matrix} ⎤ ⎥ ⎦ ⎫ ⎪ ⎬ ⎪ ⎭

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

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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