Linear Algebra Examples

⎡ ⎢ ⎣ \begin{matrix} 10 - 1 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 \sqrt{2} 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 \\ \sqrt{2} \\ 1 \end{array}]$ ,

⎡ ⎢ ⎣ \begin{matrix} 1 - \sqrt{2} 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 \\ - \sqrt{2} \\ 1 \end{array}]$

Step 1

Two vectors are orthogonal if the dot product of them is

0

$0$ .

Step 2

Evaluate the dot product of

⎡ ⎢ ⎣ \begin{matrix} 10 - 1 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 \sqrt{2} 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 \\ \sqrt{2} \\ 1 \end{array}]$ .

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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 + 0 \sqrt{2} - 1 \cdot 1

$1 \cdot 1 + 0 \sqrt{2} - 1 \cdot 1$

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

$1$ by

1

$1$ .

1 + 0 \sqrt{2} - 1 \cdot 1

$1 + 0 \sqrt{2} - 1 \cdot 1$

Step 2.2.1.2

Multiply

0

$0$ by

\sqrt{2}

$\sqrt{2}$ .

1 + 0 - 1 \cdot 1

$1 + 0 - 1 \cdot 1$

Step 2.2.1.3

Multiply

- 1

$- 1$ by

1

$1$ .

1 + 0 - 1

$1 + 0 - 1$

1 + 0 - 1

$1 + 0 - 1$

Step 2.2.2

Add

1

$1$ and

0

$0$ .

1 - 1

$1 - 1$

Step 2.2.3

Subtract

1

$1$ from

1

$1$ .

0

$0$

0

$0$

0

$0$

Step 3

Evaluate the dot product of

⎡ ⎢ ⎣ \begin{matrix} 10 - 1 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 - \sqrt{2} 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 \\ - \sqrt{2} \\ 1 \end{array}]$ .

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Step 3.1

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

1 \cdot 1 + 0 (- \sqrt{2}) - 1 \cdot 1

$1 \cdot 1 + 0 (- \sqrt{2}) - 1 \cdot 1$

Step 3.2

Simplify.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply

1

$1$ by

1

$1$ .

1 + 0 (- \sqrt{2}) - 1 \cdot 1

$1 + 0 (- \sqrt{2}) - 1 \cdot 1$

Step 3.2.1.2

Multiply

0 (- \sqrt{2})

$0 (- \sqrt{2})$ .

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Step 3.2.1.2.1

Multiply

- 1

$- 1$ by

0

$0$ .

1 + 0 \sqrt{2} - 1 \cdot 1

$1 + 0 \sqrt{2} - 1 \cdot 1$

Step 3.2.1.2.2

Multiply

0

$0$ by

\sqrt{2}

$\sqrt{2}$ .

1 + 0 - 1 \cdot 1

$1 + 0 - 1 \cdot 1$

1 + 0 - 1 \cdot 1

$1 + 0 - 1 \cdot 1$

Step 3.2.1.3

Multiply

- 1

$- 1$ by

1

$1$ .

1 + 0 - 1

$1 + 0 - 1$

1 + 0 - 1

$1 + 0 - 1$

Step 3.2.2

Add

1

$1$ and

0

$0$ .

1 - 1

$1 - 1$

Step 3.2.3

Subtract

1

$1$ from

1

$1$ .

0

$0$

0

$0$

0

$0$

Step 4

Evaluate the dot product of

⎡ ⎢ ⎣ \begin{matrix} 1 \sqrt{2} 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 \\ \sqrt{2} \\ 1 \end{array}]$ and

⎡ ⎢ ⎣ \begin{matrix} 1 - \sqrt{2} 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 \\ - \sqrt{2} \\ 1 \end{array}]$ .

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Step 4.1

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

1 \cdot 1 + \sqrt{2} (- \sqrt{2}) + 1 \cdot 1

$1 \cdot 1 + \sqrt{2} (- \sqrt{2}) + 1 \cdot 1$

Step 4.2

Simplify.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply

1

$1$ by

1

$1$ .

1 + \sqrt{2} (- \sqrt{2}) + 1 \cdot 1

$1 + \sqrt{2} (- \sqrt{2}) + 1 \cdot 1$

Step 4.2.1.2

Multiply

\sqrt{2} (- \sqrt{2})

$\sqrt{2} (- \sqrt{2})$ .

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Step 4.2.1.2.1

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

1 - ({\sqrt{2}}^{1} \sqrt{2}) + 1 \cdot 1

$1 - ({\sqrt{2}}^{1} \sqrt{2}) + 1 \cdot 1$

Step 4.2.1.2.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

1 - ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 1 \cdot 1

$1 - ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 1 \cdot 1$

Step 4.2.1.2.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

1 - {\sqrt{2}}^{1 + 1} + 1 \cdot 1

$1 - {\sqrt{2}}^{1 + 1} + 1 \cdot 1$

Step 4.2.1.2.4

Add

1

$1$ and

1

$1$ .

1 - {\sqrt{2}}^{2} + 1 \cdot 1

$1 - {\sqrt{2}}^{2} + 1 \cdot 1$

1 - {\sqrt{2}}^{2} + 1 \cdot 1

$1 - {\sqrt{2}}^{2} + 1 \cdot 1$

Step 4.2.1.3

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 4.2.1.3.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

1 - {(2^{\frac{1}{2}})}^{2} + 1 \cdot 1

$1 - {(2^{\frac{1}{2}})}^{2} + 1 \cdot 1$

Step 4.2.1.3.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

1 - 2^{\frac{1}{2} \cdot 2} + 1 \cdot 1

$1 - 2^{\frac{1}{2} \cdot 2} + 1 \cdot 1$

Step 4.2.1.3.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

1 - 2^{\frac{2}{2}} + 1 \cdot 1

$1 - 2^{\frac{2}{2}} + 1 \cdot 1$

Step 4.2.1.3.4

Cancel the common factor of

2

$2$ .

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Step 4.2.1.3.4.1

Cancel the common factor.

$1 - 2^{\frac{2}{2}} + 1 \cdot 1$

Step 4.2.1.3.4.2

Rewrite the expression.

$1 - 2^{1} + 1 \cdot 1$

Step 4.2.1.3.5

Evaluate the exponent.

$1 - 1 \cdot 2 + 1 \cdot 1$

Step 4.2.1.4

Multiply

$- 1$ by

$2$ .

$1 - 2 + 1 \cdot 1$

Step 4.2.1.5

Multiply

$1$ by

$1$ .

$1 - 2 + 1$

Step 4.2.2

Subtract

$2$ from

$1$ .

$- 1 + 1$

Step 4.2.3

Add

$- 1$ and

$1$ .

$0$

Step 5

The vectors are orthogonal since the dot products are all

$0$ .

Orthogonal

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