Find the Angle Between the Vectors (9,-2) , (-5,-6)

$(9, - 2)$ , $(- 5, - 6)$

Step 1

The equation for finding the angle between two vectors $θ$ states that the dot product of the two vectors equals the product of the magnitudes of the vectors and the cosine of the angle between them.

$u \cdot v = | u | | v | c o s (θ)$

Step 2

Solve the equation for $θ$ .

$θ = a r c \cdot c o s (\frac{u \cdot v}{| u | | v |})$

Step 3

Find the dot product of the vectors.

To find the dot product, find the sum of the products of corresponding components of the vectors.

$u \cdot v = u_{1} v_{1} + u_{2} v_{2}$

Substitute the components of the vectors into the expression.

$9 \cdot - 5 - 2 \cdot - 6$

Simplify.

Remove parentheses.

$9 \cdot - 5 - 2 \cdot - 6$

Simplify each term.

Multiply $9$ by $- 5$ .

$- 45 - 2 \cdot - 6$

Multiply $- 2$ by $- 6$ .

$- 45 + 12$

Add $- 45$ and $12$ .

$- 33$

Step 4

Find the magnitude of $u$ .

To find the magnitude of the vector, find the square root of the sum of the components of the vector squared.

$\sqrt{{(u_{1})}^{2} + {(u_{2})}^{2}}$

Substitute the components of the vector into the expression.

$\sqrt{{(9)}^{2} + {(- 2)}^{2}}$

Simplify.

Raise $9$ to the power of $2$ .

$\sqrt{81 + {(- 2)}^{2}}$

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

$\sqrt{81 + 4}$

Add $81$ and $4$ .

$\sqrt{85}$

Step 5

Find the magnitude of $v$ .

To find the magnitude of the vector, find the square root of the sum of the components of the vector squared.

$\sqrt{{(u_{1})}^{2} + {(u_{2})}^{2}}$

Substitute the components of the vector into the expression.

$\sqrt{{(- 5)}^{2} + {(- 6)}^{2}}$

Simplify.

Raise $- 5$ to the power of $2$ .

$\sqrt{25 + {(- 6)}^{2}}$

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

$\sqrt{25 + 36}$

Add $25$ and $36$ .

$\sqrt{61}$

Step 6

Substitute the values into the equation for the angle between the vectors.

$θ = \arccos (\frac{- 33}{(\sqrt{85}) \cdot (\sqrt{61})})$

Step 7

Simplify.

Simplify the denominator.

Combine using the product rule for radicals.

$\arccos (\frac{- 33}{\sqrt{85 \cdot 61}})$

Multiply $85$ by $61$ .

$\arccos (\frac{- 33}{\sqrt{5185}})$

Move the negative in front of the fraction.

$\arccos (- \frac{33}{\sqrt{5185}})$

Multiply $\frac{33}{\sqrt{5185}}$ by $\frac{\sqrt{5185}}{\sqrt{5185}}$ .

$\arccos (- (\frac{33}{\sqrt{5185}} \cdot \frac{\sqrt{5185}}{\sqrt{5185}}))$

Combine and simplify the denominator.

Multiply $\frac{33}{\sqrt{5185}}$ by $\frac{\sqrt{5185}}{\sqrt{5185}}$ .

$\arccos (- \frac{33 \sqrt{5185}}{\sqrt{5185} \sqrt{5185}})$

Raise $\sqrt{5185}$ to the power of $1$ .

$\arccos (- \frac{33 \sqrt{5185}}{{\sqrt{5185}}^{1} \sqrt{5185}})$

Raise $\sqrt{5185}$ to the power of $1$ .

$\arccos (- \frac{33 \sqrt{5185}}{{\sqrt{5185}}^{1} {\sqrt{5185}}^{1}})$

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

$\arccos (- \frac{33 \sqrt{5185}}{{\sqrt{5185}}^{1 + 1}})$

Add $1$ and $1$ .

$\arccos (- \frac{33 \sqrt{5185}}{{\sqrt{5185}}^{2}})$

Rewrite ${\sqrt{5185}}^{2}$ as $5185$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5185}$ as $5185^{\frac{1}{2}}$ .

$\arccos (- \frac{33 \sqrt{5185}}{{(5185^{\frac{1}{2}})}^{2}})$

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

$\arccos (- \frac{33 \sqrt{5185}}{5185^{\frac{1}{2} \cdot 2}})$

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

$\arccos (- \frac{33 \sqrt{5185}}{5185^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\arccos (- \frac{33 \sqrt{5185}}{5185^{\frac{2}{2}}})$

Rewrite the expression.

$\arccos (- \frac{33 \sqrt{5185}}{5185^{1}})$

Evaluate the exponent.

$\arccos (- \frac{33 \sqrt{5185}}{5185})$

Evaluate $\arccos (- \frac{33 \sqrt{5185}}{5185})$ .

$2.04686565$

Find the Angle Between the Vectors (9,-2) , (-5,-6)

Related Questions