Trigonometry Examples

$(1, - 2)$ , $(- 6, - 3)$

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.

$a⃗ \cdot b⃗ = 1 \cdot - 6 - 2 \cdot - 3$

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 $- 6$ by $1$ .

$a⃗ \cdot b⃗ = - 6 - 2 \cdot - 3$

Step 2.2.1.2

Multiply $- 2$ by $- 3$ .

$a⃗ \cdot b⃗ = - 6 + 6$

Step 2.2.2

Add $- 6$ and $6$ .

$a⃗ \cdot b⃗ = 0$

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.

$| b⃗ | = \sqrt{{(- 6)}^{2} + {(- 3)}^{2}}$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Raise $- 6$ to the power of $2$ .

$| b⃗ | = \sqrt{36 + {(- 3)}^{2}}$

Step 4.2.2

Raise $- 3$ to the power of $2$ .

$| b⃗ | = \sqrt{36 + 9}$

Step 4.2.3

Add $36$ and $9$ .

$| b⃗ | = \sqrt{45}$

Step 4.2.4

Rewrite $45$ as $3^{2} \cdot 5$ .

Tap for more steps...

Step 4.2.4.1

Factor $9$ out of $45$ .

$| b⃗ | = \sqrt{9 (5)}$

Step 4.2.4.2

Rewrite $9$ as $3^{2}$ .

$| b⃗ | = \sqrt{3^{2} \cdot 5}$

Step 4.2.5

Pull terms out from under the radical.

$| b⃗ | = 3 \sqrt{5}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{0}{\sqrt{5} (3 \sqrt{5})})$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 6.1.1

Factor $3$ out of $0$ .

$θ = \arccos (\frac{3 (0)}{\sqrt{5} (3 \sqrt{5})})$

Step 6.1.2

Cancel the common factors.

Tap for more steps...

Step 6.1.2.1

Factor $3$ out of $\sqrt{5} (3 \sqrt{5})$ .

$θ = \arccos (\frac{3 (0)}{3 (\sqrt{5} (\sqrt{5}))})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{3 \cdot 0}{3 (\sqrt{5} (\sqrt{5}))})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{0}{\sqrt{5} (\sqrt{5})})$

Step 6.2

Simplify the denominator.

Tap for more steps...

Step 6.2.1

Raise $\sqrt{5}$ to the power of $1$ .

$θ = \arccos (\frac{0}{{\sqrt{5}}^{1} \sqrt{5}})$

Step 6.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$θ = \arccos (\frac{0}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}})$

Step 6.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$θ = \arccos (\frac{0}{{\sqrt{5}}^{1 + 1}})$

Step 6.2.4

Add $1$ and $1$ .

$θ = \arccos (\frac{0}{{\sqrt{5}}^{2}})$

Step 6.3

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$θ = \arccos (\frac{0}{{(5^{\frac{1}{2}})}^{2}})$

Step 6.3.2

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

$θ = \arccos (\frac{0}{5^{\frac{1}{2} \cdot 2}})$

Step 6.3.3

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

$θ = \arccos (\frac{0}{5^{\frac{2}{2}}})$

Step 6.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.4.1

Cancel the common factor.

$θ = \arccos (\frac{0}{5^{\frac{2}{2}}})$

Step 6.3.4.2

Rewrite the expression.

$θ = \arccos (\frac{0}{5^{1}})$

Step 6.3.5

Evaluate the exponent.

$θ = \arccos (\frac{0}{5})$

Step 6.4

Divide $0$ by $5$ .

$θ = \arccos (0)$

Step 6.5

The exact value of $\arccos (0)$ is $90$ .

$θ = 90$