Linear Algebra Examples

$a = [\begin{array}{c} 2 & 3 & 2 \end{array}]$ , $b = [\begin{array}{c} 1 & 2 & 1 \end{array}]$

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.

$a⃗ \cdot b⃗ = 2 \cdot 1 + 3 \cdot 2 + 2 \cdot 1$

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 $2$ by $1$ .

$a⃗ \cdot b⃗ = 2 + 3 \cdot 2 + 2 \cdot 1$

Step 1.2.1.2

Multiply $3$ by $2$ .

$a⃗ \cdot b⃗ = 2 + 6 + 2 \cdot 1$

Step 1.2.1.3

Multiply $2$ by $1$ .

$a⃗ \cdot b⃗ = 2 + 6 + 2$

Step 1.2.2

Add $2$ and $6$ .

$a⃗ \cdot b⃗ = 8 + 2$

Step 1.2.3

Add $8$ and $2$ .

$a⃗ \cdot b⃗ = 10$

Step 2

Find the norm of $b⃗ = [\begin{array}{c} 1 & 2 & 1 \end{array}]$ .

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Step 2.1

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

$| | b⃗ | | = \sqrt{1^{2} + 2^{2} + 1^{2}}$

Step 2.2

Simplify.

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Step 2.2.1

One to any power is one.

$| | b⃗ | | = \sqrt{1 + 2^{2} + 1^{2}}$

Step 2.2.2

Raise $2$ to the power of $2$ .

$| | b⃗ | | = \sqrt{1 + 4 + 1^{2}}$

Step 2.2.3

One to any power is one.

$| | b⃗ | | = \sqrt{1 + 4 + 1}$

Step 2.2.4

Add $1$ and $4$ .

$| | b⃗ | | = \sqrt{5 + 1}$

Step 2.2.5

Add $5$ and $1$ .

$| | b⃗ | | = \sqrt{6}$

Step 3

Find the projection of $a⃗$ onto $b⃗$ using the projection formula.

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

Step 4

Substitute $10$ for $a⃗ \cdot b⃗$ .

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

Step 5

Substitute $\sqrt{6}$ for $| | b⃗ | |$ .

${proj}_{b⃗} (a⃗) = \frac{10}{{\sqrt{6}}^{2}} \times b⃗$

Step 6

Substitute $[\begin{array}{c} 1 & 2 & 1 \end{array}]$ for $b⃗$ .

${proj}_{b⃗} (a⃗) = \frac{10}{{\sqrt{6}}^{2}} \times [\begin{array}{c} 1 & 2 & 1 \end{array}]$

Step 7

Simplify the right side.

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Step 7.1

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

${proj}_{b⃗} (a⃗) = \frac{10}{{(6^{\frac{1}{2}})}^{2}} \times [\begin{array}{c} 1 & 2 & 1 \end{array}]$

Step 7.1.2

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

${proj}_{b⃗} (a⃗) = \frac{10}{6^{\frac{1}{2} \cdot 2}} \times [\begin{array}{c} 1 & 2 & 1 \end{array}]$

Step 7.1.3

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

${proj}_{b⃗} (a⃗) = \frac{10}{6^{\frac{2}{2}}} \times [\begin{array}{c} 1 & 2 & 1 \end{array}]$

Step 7.1.4

Cancel the common factor of $2$ .

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Step 7.1.4.1

Cancel the common factor.

${proj}_{b⃗} (a⃗) = \frac{10}{6^{\frac{2}{2}}} \times [\begin{array}{c} 1 & 2 & 1 \end{array}]$

Step 7.1.4.2

Rewrite the expression.

${proj}_{b⃗} (a⃗) = \frac{10}{6^{1}} \times [\begin{array}{c} 1 & 2 & 1 \end{array}]$

Step 7.1.5

Evaluate the exponent.

${proj}_{b⃗} (a⃗) = \frac{10}{6} \times [\begin{array}{c} 1 & 2 & 1 \end{array}]$

Step 7.2

Cancel the common factor of $10$ and $6$ .

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Step 7.2.1

Factor $2$ out of $10$ .

${proj}_{b⃗} (a⃗) = \frac{2 (5)}{6} \times [\begin{array}{c} 1 & 2 & 1 \end{array}]$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $2$ out of $6$ .

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

Step 7.2.2.2

Cancel the common factor.

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

Step 7.2.2.3

Rewrite the expression.

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

Step 7.3

Multiply $\frac{5}{3}$ by each element of the matrix.

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

Step 7.4

Simplify each element in the matrix.

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Step 7.4.1

Multiply $\frac{5}{3}$ by $1$ .

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

Step 7.4.2

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

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Step 7.4.2.1

Combine $\frac{5}{3}$ and $2$ .

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

Step 7.4.2.2

Multiply $5$ by $2$ .

${proj}_{b⃗} (a⃗) = [\begin{array}{c} \frac{5}{3} & \frac{10}{3} & \frac{5}{3} \cdot 1 \end{array}]$

Step 7.4.3

Multiply $\frac{5}{3}$ by $1$ .

${proj}_{b⃗} (a⃗) = [\begin{array}{c} \frac{5}{3} & \frac{10}{3} & \frac{5}{3} \end{array}]$

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