Linear Algebra Examples

$[\begin{array}{c} 1 \\ 0 \\ - 1 \end{array}]$ , $[\begin{array}{c} 1 \\ \sqrt{2} \\ 1 \end{array}]$ , $[\begin{array}{c} 1 \\ - \sqrt{2} \\ 1 \end{array}]$

Step 1

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

Step 2

Evaluate the dot product of $[\begin{array}{c} 1 \\ 0 \\ - 1 \end{array}]$ and $[\begin{array}{c} 1 \\ \sqrt{2} \\ 1 \end{array}]$ .

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Step 2.1

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

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

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 $1$ .

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

Step 2.2.1.2

Multiply $0$ by $\sqrt{2}$ .

$1 + 0 - 1 \cdot 1$

Step 2.2.1.3

Multiply $- 1$ by $1$ .

$1 + 0 - 1$

Step 2.2.2

Add $1$ and $0$ .

$1 - 1$

Step 2.2.3

Subtract $1$ from $1$ .

$0$

Step 3

Evaluate the dot product of $[\begin{array}{c} 1 \\ 0 \\ - 1 \end{array}]$ and $[\begin{array}{c} 1 \\ - \sqrt{2} \\ 1 \end{array}]$ .

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Step 3.1

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

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

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 $1$ by $1$ .

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

Multiply $- 1$ by $0$ .

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

Step 3.2.1.2.2

Multiply $0$ by $\sqrt{2}$ .

$1 + 0 - 1 \cdot 1$

Step 3.2.1.3

Multiply $- 1$ by $1$ .

$1 + 0 - 1$

Step 3.2.2

Add $1$ and $0$ .

$1 - 1$

Step 3.2.3

Subtract $1$ from $1$ .

$0$

Step 4

Evaluate the dot product of $[\begin{array}{c} 1 \\ \sqrt{2} \\ 1 \end{array}]$ and $[\begin{array}{c} 1 \\ - \sqrt{2} \\ 1 \end{array}]$ .

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Step 4.1

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

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

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 $1$ by $1$ .

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

Step 4.2.1.2

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

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Step 4.2.1.2.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.2.1.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.2.1.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2.1.2.4

Add $1$ and $1$ .

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

Step 4.2.1.3

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

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Step 4.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 4.2.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.1.3.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 4.2.1.3.4

Cancel the common factor of $2$ .

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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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