Precalculus Examples

$(1, 2)$ ,

$(2, 9)$

Step 1

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

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

Step 2

Find the dot product.

Tap for more steps...

Step 2.1

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

$a⃗ \cdot b⃗ = 1 \cdot 2 + 2 \cdot 9$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Multiply

$2$ by

$1$ .

$a⃗ \cdot b⃗ = 2 + 2 \cdot 9$

Step 2.2.1.2

Multiply

$2$ by

$9$ .

$a⃗ \cdot b⃗ = 2 + 18$

Step 2.2.2

Add

$2$ and

$18$ .

$a⃗ \cdot b⃗ = 20$

Step 3

Find the magnitude of

$a⃗$ .

Tap for more steps...

Step 3.1

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

$| a⃗ | = \sqrt{1^{2} + 2^{2}}$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

One to any power is one.

$| a⃗ | = \sqrt{1 + 2^{2}}$

Step 3.2.2

Raise

$2$ to the power of

$2$ .

$| a⃗ | = \sqrt{1 + 4}$

Step 3.2.3

Add

$1$ and

$4$ .

$| a⃗ | = \sqrt{5}$

Step 4

Find the magnitude of

$b⃗$ .

Tap for more steps...

Step 4.1

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

$| b⃗ | = \sqrt{2^{2} + 9^{2}}$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Raise

$2$ to the power of

$2$ .

$| b⃗ | = \sqrt{4 + 9^{2}}$

Step 4.2.2

Raise

$9$ to the power of

$2$ .

$| b⃗ | = \sqrt{4 + 81}$

Step 4.2.3

Add

$4$ and

$81$ .

$| b⃗ | = \sqrt{85}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{20}{\sqrt{5} \sqrt{85}})$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Simplify the denominator.

Tap for more steps...

Step 6.1.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{20}{\sqrt{5 \cdot 85}})$

Step 6.1.2

Multiply

$5$ by

$85$ .

$θ = \arccos (\frac{20}{\sqrt{425}})$

Step 6.2

Simplify the denominator.

Tap for more steps...

Step 6.2.1

Rewrite

$425$ as

$5^{2} \cdot 17$ .

Tap for more steps...

Step 6.2.1.1

Factor

$25$ out of

$425$ .

$θ = \arccos (\frac{20}{\sqrt{25 (17)}})$

Step 6.2.1.2

Rewrite

$25$ as

$5^{2}$ .

$θ = \arccos (\frac{20}{\sqrt{5^{2} \cdot 17}})$

Step 6.2.2

Pull terms out from under the radical.

$θ = \arccos (\frac{20}{5 \sqrt{17}})$

Step 6.3

Cancel the common factor of

$20$ and

$5$ .

Tap for more steps...

Step 6.3.1

Factor

$5$ out of

$20$ .

$θ = \arccos (\frac{5 \cdot 4}{5 \sqrt{17}})$

Step 6.3.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.1

Factor

$5$ out of

$5 \sqrt{17}$ .

$θ = \arccos (\frac{5 \cdot 4}{5 (\sqrt{17})})$

Step 6.3.2.2

Cancel the common factor.

$θ = \arccos (\frac{5 \cdot 4}{5 \sqrt{17}})$

Step 6.3.2.3

Rewrite the expression.

$θ = \arccos (\frac{4}{\sqrt{17}})$

Step 6.4

Multiply

$\frac{4}{\sqrt{17}}$ by

$\frac{\sqrt{17}}{\sqrt{17}}$ .

$θ = \arccos (\frac{4}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}})$

Step 6.5

Combine and simplify the denominator.

Tap for more steps...

Step 6.5.1

Multiply

$\frac{4}{\sqrt{17}}$ by

$\frac{\sqrt{17}}{\sqrt{17}}$ .

$θ = \arccos (\frac{4 \sqrt{17}}{\sqrt{17} \sqrt{17}})$

Step 6.5.2

Raise

$\sqrt{17}$ to the power of

$1$ .

$θ = \arccos (\frac{4 \sqrt{17}}{{\sqrt{17}}^{1} \sqrt{17}})$

Step 6.5.3

Raise

$\sqrt{17}$ to the power of

$1$ .

$θ = \arccos (\frac{4 \sqrt{17}}{{\sqrt{17}}^{1} {\sqrt{17}}^{1}})$

Step 6.5.4

Use the power rule

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

$θ = \arccos (\frac{4 \sqrt{17}}{{\sqrt{17}}^{1 + 1}})$

Step 6.5.5

Add

$1$ and

$1$ .

$θ = \arccos (\frac{4 \sqrt{17}}{{\sqrt{17}}^{2}})$

Step 6.5.6

Rewrite

${\sqrt{17}}^{2}$ as

$17$ .

Tap for more steps...

Step 6.5.6.1

Use

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

$\sqrt{17}$ as

$17^{\frac{1}{2}}$ .

$θ = \arccos (\frac{4 \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}})$

Step 6.5.6.2

Apply the power rule and multiply exponents,

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

$θ = \arccos (\frac{4 \sqrt{17}}{17^{\frac{1}{2} \cdot 2}})$

Step 6.5.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$θ = \arccos (\frac{4 \sqrt{17}}{17^{\frac{2}{2}}})$

Step 6.5.6.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 6.5.6.4.1

Cancel the common factor.

$θ = \arccos (\frac{4 \sqrt{17}}{17^{\frac{2}{2}}})$

Step 6.5.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{4 \sqrt{17}}{17^{1}})$

Step 6.5.6.5

Evaluate the exponent.

$θ = \arccos (\frac{4 \sqrt{17}}{17})$

Step 6.6

Evaluate

$\arccos (\frac{4 \sqrt{17}}{17})$ .

$θ = 14.03624346$

Enter YOUR Problem