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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