Finite Math Examples

(9, 8)

$(9, 8)$ ,

(9, 3)

$(9, 3)$

Step 1

Use the dot product formula to find the angle between two vectors.

θ = arccos (\frac{a⃗ \cdot b⃗}{| a⃗ | | b⃗ |})

$θ = \arccos (\frac{a⃗ \cdot b⃗}{| a⃗ | | b⃗ |})$

Step 2

Find the dot product.

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Step 2.1

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

a⃗ \cdot b⃗ = 9 \cdot 9 + 8 \cdot 3

$a⃗ \cdot b⃗ = 9 \cdot 9 + 8 \cdot 3$

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

9

$9$ by

9

$9$ .

a⃗ \cdot b⃗ = 81 + 8 \cdot 3

$a⃗ \cdot b⃗ = 81 + 8 \cdot 3$

Step 2.2.1.2

Multiply

8

$8$ by

3

$3$ .

a⃗ \cdot b⃗ = 81 + 24

$a⃗ \cdot b⃗ = 81 + 24$

a⃗ \cdot b⃗ = 81 + 24

$a⃗ \cdot b⃗ = 81 + 24$

Step 2.2.2

Add

81

$81$ and

24

$24$ .

a⃗ \cdot b⃗ = 105

$a⃗ \cdot b⃗ = 105$

a⃗ \cdot b⃗ = 105

$a⃗ \cdot b⃗ = 105$

a⃗ \cdot b⃗ = 105

$a⃗ \cdot b⃗ = 105$

Step 3

Find the magnitude of

a⃗

$a⃗$ .

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Step 3.1

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

| a⃗ | = \sqrt{9^{2} + 8^{2}}

$| a⃗ | = \sqrt{9^{2} + 8^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raise

9

$9$ to the power of

2

$2$ .

| a⃗ | = \sqrt{81 + 8^{2}}

$| a⃗ | = \sqrt{81 + 8^{2}}$

Step 3.2.2

Raise

8

$8$ to the power of

2

$2$ .

| a⃗ | = \sqrt{81 + 64}

$| a⃗ | = \sqrt{81 + 64}$

Step 3.2.3

Add

81

$81$ and

64

$64$ .

| a⃗ | = \sqrt{145}

$| a⃗ | = \sqrt{145}$

| a⃗ | = \sqrt{145}

$| a⃗ | = \sqrt{145}$

| a⃗ | = \sqrt{145}

$| a⃗ | = \sqrt{145}$

Step 4

Find the magnitude of

b⃗

$b⃗$ .

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Step 4.1

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

| b⃗ | = \sqrt{9^{2} + 3^{2}}

$| b⃗ | = \sqrt{9^{2} + 3^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

Raise

9

$9$ to the power of

2

$2$ .

| b⃗ | = \sqrt{81 + 3^{2}}

$| b⃗ | = \sqrt{81 + 3^{2}}$

Step 4.2.2

Raise

3

$3$ to the power of

2

$2$ .

| b⃗ | = \sqrt{81 + 9}

$| b⃗ | = \sqrt{81 + 9}$

Step 4.2.3

Add

81

$81$ and

9

$9$ .

| b⃗ | = \sqrt{90}

$| b⃗ | = \sqrt{90}$

Step 4.2.4

Rewrite

90

$90$ as

3^{2} \cdot 10

$3^{2} \cdot 10$ .

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Step 4.2.4.1

Factor

9

$9$ out of

90

$90$ .

| b⃗ | = \sqrt{9 (10)}

$| b⃗ | = \sqrt{9 (10)}$

Step 4.2.4.2

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

| b⃗ | = \sqrt{3^{2} \cdot 10}

$| b⃗ | = \sqrt{3^{2} \cdot 10}$

| b⃗ | = \sqrt{3^{2} \cdot 10}

$| b⃗ | = \sqrt{3^{2} \cdot 10}$

Step 4.2.5

Pull terms out from under the radical.

| b⃗ | = 3 \sqrt{10}

$| b⃗ | = 3 \sqrt{10}$

| b⃗ | = 3 \sqrt{10}

$| b⃗ | = 3 \sqrt{10}$

| b⃗ | = 3 \sqrt{10}

$| b⃗ | = 3 \sqrt{10}$

Step 5

Substitute the values into the formula.

θ = arccos ⎛ ⎜ ⎝ \frac{105}{\sqrt{145} (3 \sqrt{10})} ⎞ ⎟ ⎠

$θ = \arccos (\frac{105}{\sqrt{145} (3 \sqrt{10})})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of

105

$105$ and

3

$3$ .

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Step 6.1.1

Factor

3

$3$ out of

105

$105$ .

θ = arccos ⎛ ⎜ ⎝ \frac{3 \cdot 35}{\sqrt{145} (3 \sqrt{10})} ⎞ ⎟ ⎠

$θ = \arccos (\frac{3 \cdot 35}{\sqrt{145} (3 \sqrt{10})})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor

3

$3$ out of

\sqrt{145} (3 \sqrt{10})

$\sqrt{145} (3 \sqrt{10})$ .

θ = arccos ⎛ ⎜ ⎝ \frac{3 \cdot 35}{3 (\sqrt{145} (\sqrt{10}))} ⎞ ⎟ ⎠

$θ = \arccos (\frac{3 \cdot 35}{3 (\sqrt{145} (\sqrt{10}))})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{3 \cdot 35}{3 (\sqrt{145} (\sqrt{10}))})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{35}{\sqrt{145} (\sqrt{10})})$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{35}{\sqrt{145 \cdot 10}})$

Step 6.2.2

Multiply

$145$ by

$10$ .

$θ = \arccos (\frac{35}{\sqrt{1450}})$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Rewrite

$1450$ as

$5^{2} \cdot 58$ .

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Step 6.3.1.1

Factor

$25$ out of

$1450$ .

$θ = \arccos (\frac{35}{\sqrt{25 (58)}})$

Step 6.3.1.2

Rewrite

$25$ as

$5^{2}$ .

$θ = \arccos (\frac{35}{\sqrt{5^{2} \cdot 58}})$

Step 6.3.2

Pull terms out from under the radical.

$θ = \arccos (\frac{35}{5 \sqrt{58}})$

Step 6.4

Cancel the common factor of

$35$ and

$5$ .

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Step 6.4.1

Factor

$5$ out of

$35$ .

$θ = \arccos (\frac{5 \cdot 7}{5 \sqrt{58}})$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor

$5$ out of

$5 \sqrt{58}$ .

$θ = \arccos (\frac{5 \cdot 7}{5 (\sqrt{58})})$

Step 6.4.2.2

Cancel the common factor.

$θ = \arccos (\frac{5 \cdot 7}{5 \sqrt{58}})$

Step 6.4.2.3

Rewrite the expression.

$θ = \arccos (\frac{7}{\sqrt{58}})$

Step 6.5

Multiply

$\frac{7}{\sqrt{58}}$ by

$\frac{\sqrt{58}}{\sqrt{58}}$ .

$θ = \arccos (\frac{7}{\sqrt{58}} \cdot \frac{\sqrt{58}}{\sqrt{58}})$

Step 6.6

Combine and simplify the denominator.

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Step 6.6.1

Multiply

$\frac{7}{\sqrt{58}}$ by

$\frac{\sqrt{58}}{\sqrt{58}}$ .

$θ = \arccos (\frac{7 \sqrt{58}}{\sqrt{58} \sqrt{58}})$

Step 6.6.2

Raise

$\sqrt{58}$ to the power of

$1$ .

$θ = \arccos (\frac{7 \sqrt{58}}{{\sqrt{58}}^{1} \sqrt{58}})$

Step 6.6.3

Raise

$\sqrt{58}$ to the power of

$1$ .

$θ = \arccos (\frac{7 \sqrt{58}}{{\sqrt{58}}^{1} {\sqrt{58}}^{1}})$

Step 6.6.4

Use the power rule

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

$θ = \arccos (\frac{7 \sqrt{58}}{{\sqrt{58}}^{1 + 1}})$

Step 6.6.5

Add

$1$ and

$1$ .

$θ = \arccos (\frac{7 \sqrt{58}}{{\sqrt{58}}^{2}})$

Step 6.6.6

Rewrite

${\sqrt{58}}^{2}$ as

$58$ .

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Step 6.6.6.1

Use

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

$\sqrt{58}$ as

$58^{\frac{1}{2}}$ .

$θ = \arccos (\frac{7 \sqrt{58}}{{(58^{\frac{1}{2}})}^{2}})$

Step 6.6.6.2

Apply the power rule and multiply exponents,

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

$θ = \arccos (\frac{7 \sqrt{58}}{58^{\frac{1}{2} \cdot 2}})$

Step 6.6.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$θ = \arccos (\frac{7 \sqrt{58}}{58^{\frac{2}{2}}})$

Step 6.6.6.4

Cancel the common factor of

$2$ .

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Step 6.6.6.4.1

Cancel the common factor.

$θ = \arccos (\frac{7 \sqrt{58}}{58^{\frac{2}{2}}})$

Step 6.6.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{7 \sqrt{58}}{58^{1}})$

Step 6.6.6.5

Evaluate the exponent.

$θ = \arccos (\frac{7 \sqrt{58}}{58})$

Step 6.7

Evaluate

$\arccos (\frac{7 \sqrt{58}}{58})$ .

$θ = 23.19859051$