Trigonometry Examples

$(- 3, 0)$ , $(- 3, - 4)$

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.

$a⃗ \cdot b⃗ = - 3 \cdot - 3 + 0 \cdot - 4$

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 $- 3$ by $- 3$ .

$a⃗ \cdot b⃗ = 9 + 0 \cdot - 4$

Step 2.2.1.2

Multiply $0$ by $- 4$ .

$a⃗ \cdot b⃗ = 9 + 0$

Step 2.2.2

Add $9$ and $0$ .

$a⃗ \cdot b⃗ = 9$

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.

$| a⃗ | = \sqrt{{(- 3)}^{2} + 0^{2}}$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

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

$| a⃗ | = \sqrt{9 + 0^{2}}$

Step 3.2.2

Raising $0$ to any positive power yields $0$ .

$| a⃗ | = \sqrt{9 + 0}$

Step 3.2.3

Add $9$ and $0$ .

$| a⃗ | = \sqrt{9}$

Step 3.2.4

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

$| a⃗ | = \sqrt{3^{2}}$

Step 3.2.5

Pull terms out from under the radical, assuming positive real numbers.

$| a⃗ | = 3$

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.

$| b⃗ | = \sqrt{{(- 3)}^{2} + {(- 4)}^{2}}$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

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

$| b⃗ | = \sqrt{9 + {(- 4)}^{2}}$

Step 4.2.2

Raise $- 4$ to the power of $2$ .

$| b⃗ | = \sqrt{9 + 16}$

Step 4.2.3

Add $9$ and $16$ .

$| b⃗ | = \sqrt{25}$

Step 4.2.4

Rewrite $25$ as $5^{2}$ .

$| b⃗ | = \sqrt{5^{2}}$

Step 4.2.5

Pull terms out from under the radical, assuming positive real numbers.

$| b⃗ | = 5$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{9}{3 \cdot 5})$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Cancel the common factor of $9$ and $3$ .

Tap for more steps...

Step 6.1.1

Factor $3$ out of $9$ .

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

Step 6.1.2

Cancel the common factors.

Tap for more steps...

Step 6.1.2.1

Factor $3$ out of $3 \cdot 5$ .

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

Step 6.1.2.2

Cancel the common factor.

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

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{3}{5})$

Step 6.2

Evaluate $\arccos (\frac{3}{5})$ .

$θ = 53.13010235$