Linear Algebra Examples

$a = [\begin{array}{c} 1 & 0 & 3 \end{array}]$ ,

$b = [\begin{array}{c} 1 & 1 & 1 \end{array}]$

Step 1

Find the dot product.

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Step 1.1

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

$b⃗ \cdot a⃗ = 1 \cdot 1 + 1 \cdot 0 + 1 \cdot 3$

Step 1.2

Simplify.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply

$1$ by

$1$ .

$b⃗ \cdot a⃗ = 1 + 1 \cdot 0 + 1 \cdot 3$

Step 1.2.1.2

Multiply

$0$ by

$1$ .

$b⃗ \cdot a⃗ = 1 + 0 + 1 \cdot 3$

Step 1.2.1.3

Multiply

$3$ by

$1$ .

$b⃗ \cdot a⃗ = 1 + 0 + 3$

Step 1.2.2

Add

$1$ and

$0$ .

$b⃗ \cdot a⃗ = 1 + 3$

Step 1.2.3

Add

$1$ and

$3$ .

$b⃗ \cdot a⃗ = 4$

Step 2

Find the norm of

$a⃗ = [\begin{array}{c} 1 & 0 & 3 \end{array}]$ .

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Step 2.1

The norm is the square root of the sum of squares of each element in the vector.

$| | a⃗ | | = \sqrt{1^{2} + 0^{2} + 3^{2}}$

Step 2.2

Simplify.

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Step 2.2.1

One to any power is one.

$| | a⃗ | | = \sqrt{1 + 0^{2} + 3^{2}}$

Step 2.2.2

Raising

$0$ to any positive power yields

$0$ .

$| | a⃗ | | = \sqrt{1 + 0 + 3^{2}}$

Step 2.2.3

Raise

$3$ to the power of

$2$ .

$| | a⃗ | | = \sqrt{1 + 0 + 9}$

Step 2.2.4

Add

$1$ and

$0$ .

$| | a⃗ | | = \sqrt{1 + 9}$

Step 2.2.5

Add

$1$ and

$9$ .

$| | a⃗ | | = \sqrt{10}$

Step 3

Find the projection of

$b⃗$ onto

$a⃗$ using the projection formula.

${proj}_{a⃗} (b⃗) = \frac{b⃗ \cdot a⃗}{{| | a⃗ | |}^{2}} \times a⃗$

Step 4

Substitute

$4$ for

$b⃗ \cdot a⃗$ .

${proj}_{a⃗} (b⃗) = \frac{4}{{| | a⃗ | |}^{2}} \times a⃗$

Step 5

Substitute

$\sqrt{10}$ for

$| | a⃗ | |$ .

${proj}_{a⃗} (b⃗) = \frac{4}{{\sqrt{10}}^{2}} \times a⃗$

Step 6

Substitute

$[\begin{array}{c} 1 & 0 & 3 \end{array}]$ for

$a⃗$ .

${proj}_{a⃗} (b⃗) = \frac{4}{{\sqrt{10}}^{2}} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7

Simplify the right side.

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Step 7.1

Rewrite

${\sqrt{10}}^{2}$ as

$10$ .

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Step 7.1.1

Use

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

$\sqrt{10}$ as

$10^{\frac{1}{2}}$ .

${proj}_{a⃗} (b⃗) = \frac{4}{{(10^{\frac{1}{2}})}^{2}} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.1.2

Apply the power rule and multiply exponents,

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

${proj}_{a⃗} (b⃗) = \frac{4}{10^{\frac{1}{2} \cdot 2}} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.1.3

Combine

$\frac{1}{2}$ and

$2$ .

${proj}_{a⃗} (b⃗) = \frac{4}{10^{\frac{2}{2}}} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.1.4

Cancel the common factor of

$2$ .

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Step 7.1.4.1

Cancel the common factor.

${proj}_{a⃗} (b⃗) = \frac{4}{10^{\frac{2}{2}}} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.1.4.2

Rewrite the expression.

${proj}_{a⃗} (b⃗) = \frac{4}{10^{1}} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.1.5

Evaluate the exponent.

${proj}_{a⃗} (b⃗) = \frac{4}{10} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.2

Cancel the common factor of

$4$ and

$10$ .

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Step 7.2.1

Factor

$2$ out of

$4$ .

${proj}_{a⃗} (b⃗) = \frac{2 (2)}{10} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor

$2$ out of

$10$ .

${proj}_{a⃗} (b⃗) = \frac{2 \cdot 2}{2 \cdot 5} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.2.2.2

Cancel the common factor.

${proj}_{a⃗} (b⃗) = \frac{2 \cdot 2}{2 \cdot 5} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.2.2.3

Rewrite the expression.

${proj}_{a⃗} (b⃗) = \frac{2}{5} \times [\begin{array}{c} 1 & 0 & 3 \end{array}]$

Step 7.3

Multiply

$\frac{2}{5}$ by each element of the matrix.

${proj}_{a⃗} (b⃗) = [\begin{array}{c} \frac{2}{5} \cdot 1 & \frac{2}{5} \cdot 0 & \frac{2}{5} \cdot 3 \end{array}]$

Step 7.4

Simplify each element in the matrix.

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Step 7.4.1

Multiply

$\frac{2}{5}$ by

$1$ .

${proj}_{a⃗} (b⃗) = [\begin{array}{c} \frac{2}{5} & \frac{2}{5} \cdot 0 & \frac{2}{5} \cdot 3 \end{array}]$

Step 7.4.2

Multiply

$\frac{2}{5}$ by

$0$ .

${proj}_{a⃗} (b⃗) = [\begin{array}{c} \frac{2}{5} & 0 & \frac{2}{5} \cdot 3 \end{array}]$

Step 7.4.3

Multiply

$\frac{2}{5} \cdot 3$ .

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Step 7.4.3.1

Combine

$\frac{2}{5}$ and

$3$ .

${proj}_{a⃗} (b⃗) = [\begin{array}{c} \frac{2}{5} & 0 & \frac{2 \cdot 3}{5} \end{array}]$

Step 7.4.3.2

Multiply

$2$ by

$3$ .

${proj}_{a⃗} (b⃗) = [\begin{array}{c} \frac{2}{5} & 0 & \frac{6}{5} \end{array}]$

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