Find the Angle Between the Vectors (5,0) , (0,9)

$(5, 0)$ , $(0, 9)$

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.

$u \cdot v = | u | | v | c o s (θ)$

Solve the equation for $θ$ .

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

Find the dot product of the vectors.

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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.

$5 \cdot 0 + 0 \cdot 9$

Simplify.

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Remove parentheses.

$5 \cdot 0 + 0 \cdot 9$

Simplify each term.

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Multiply $5$ by $0$ .

$0 + 0 \cdot 9$

Multiply $0$ by $9$ .

$0 + 0$

Add $0$ and $0$ .

$0$

Find the magnitude of $u$ .

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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.

$\sqrt{{(5)}^{2} + {(0)}^{2}}$

Simplify.

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

$\sqrt{25 + {(0)}^{2}}$

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

$\sqrt{25 + 0}$

Add $25$ and $0$ .

$\sqrt{25}$

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

$\sqrt{5^{2}}$

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

$5$

Find the magnitude of $v$ .

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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.

$\sqrt{{(0)}^{2} + {(9)}^{2}}$

Simplify.

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

$\sqrt{0 + {(9)}^{2}}$

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

$\sqrt{0 + 81}$

Add $0$ and $81$ .

$\sqrt{81}$

Rewrite $81$ as $9^{2}$ .

$\sqrt{9^{2}}$

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

$9$

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

$θ = \arccos (\frac{0}{(5) \cdot (9)})$

Simplify.

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Cancel the common factor of $0$ and $5$ .

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Factor $5$ out of $0$ .

$\arccos (\frac{5 (0)}{(5) \cdot (9)})$

Cancel the common factors.

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Factor $5$ out of $(5) \cdot (9)$ .

$\arccos (\frac{5 (0)}{(5) (9)})$

Cancel the common factor.

$\arccos (\frac{5 \cdot 0}{5 \cdot 9})$

Rewrite the expression.

$\arccos (\frac{0}{9})$

Cancel the common factor of $0$ and $9$ .

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Factor $9$ out of $0$ .

$\arccos (\frac{9 (0)}{9})$

Cancel the common factors.

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Factor $9$ out of $9$ .

$\arccos (\frac{9 \cdot 0}{9 \cdot 1})$

Cancel the common factor.

$\arccos (\frac{9 \cdot 0}{9 \cdot 1})$

Rewrite the expression.

$\arccos (\frac{0}{1})$

Divide $0$ by $1$ .

$\arccos (0)$

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$\frac{π}{2}$

Find the Angle Between the Vectors (5,0) , (0,9)

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