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High School Math : Parametric, Polar, and Vector

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Example Questions

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Example Question #1 :Understanding Polar Coordinates

The polar coordinates of a point are $\left(2.1, \frac{7\pi }{3}\right )$ . Give its-coordinate in the rectangular coordinate system (nearest hundredth).

Possible Answers:

4.20

1.05

-1.05

-1.82

1.82

Correct answer:

1.82

Explanation:

Given the polar coordinates $(r,\theta )$ , the-coordinate is $y= r \sin \theta$ . We can find this coordinate by substituting $r = 2.1, \theta = \frac{7\pi }{3}$ :

$y = r \sin \theta = 2.1 \cdot \sin \frac{7\pi }{3} \approx 2.1 \cdot 0.8660 \approx 1.82$

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Example Question #2 :Calculus Ii — Integrals

The polar coordinates of a point are $\left(2.1, \frac{7\pi }{3}\right )$ . Give its-coordinate in the rectangular coordinate system (nearest hundredth).

Possible Answers:

1.05

-1.82

-1.05

1.82

4.20

Correct answer:

1.05

Explanation:

Given the polar coordinates $(r,\theta )$ , the-coordinate is $x= r \cos \theta$ . We can find this coordinate by substituting $r = 2.1, \theta = \frac{7\pi }{3}$ :

$x = r \cos \theta = 2.1 \cdot \cos \frac{7\pi }{3} = 2.1 \cdot 0.5 = 1.05$

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Example Question #3 :Understanding Polar Coordinates

The polar coordinates of a point are $\left(1.2, \frac{2\pi }{5}\right )$ . Give its-coordinate in the rectangular coordinate system (nearest hundredth).

Possible Answers:

1.14

3.69

3.88

0.37

1.26

Correct answer:

1.14

Explanation:

Given the polar coordinates $(r,\theta )$ , the-coordinate is $y= r \sin \theta$ . We can find this coordinate by substituting $r = 1.2, \theta = \frac{2\pi }{5}$ :

$y = r \cos \theta = 1.2 \cdot \sin \frac{2\pi }{5} \approx 1.2 \cdot 0.9511 \approx 1.14$

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Example Question #1 :Understanding Polar Coordinates

The polar coordinates of a point are $\left(1.2, \frac{2\pi }{5}\right )$ . Give its-coordinate in the rectangular coordinate system (nearest hundredth).

Possible Answers:

3.69

3.88

0.37

1.14

1.26

Correct answer:

0.37

Explanation:

Given the polar coordinates $(r,\theta )$ , the-coordinate is $x= r \cos \theta$ . We can find this coordinate by substituting $r = 1.2, \theta = \frac{2\pi }{5}$ :

$x = r \cos \theta = 1.2 \cdot \cos \frac{2\pi }{5} \approx 1.2 \cdot 0.3090 \approx 0.37$

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Example Question #1 :Calculus Ii — Integrals

Find the vector where its initial point is (-1,4) and its terminal point is (2,2) .

Possible Answers:

<1,2>

<1,-2>

<1,6>

<3,-2>

Correct answer:

<3,-2>

Explanation:

We need to subtract the-coordinate and the-coordinates to solve for a vector when given its initial and terminal coordinates:

Initial pt: (-1,4)

Terminal pt: (2,2)

Vector: <2-(-1),2-4>

Vector:

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Example Question #1 :Parametric, Polar, And Vector

Find the vector where its initial point is (-1,-5) and its terminal point is (2,3) .

Possible Answers:

$\left \langle1,2\right \rangle$

$\left \langle1,8\right \rangle$

$\left \langle3,2\right \rangle$

$\left \langle3,8\right \rangle$

Correct answer:

$\left \langle3,8\right \rangle$

Explanation:

We need to subtract the-coordinate and the-coordinate to solve for a vector when given its initial and terminal coordinates:

Initial pt: (-1,-5)

Terminal pt: (2,3)

Vector: $\left \langle 2-(-1),3-(-5)\right \rangle$

Vector: $\left \langle3,8\right \rangle$

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Example Question #11 :Calculus Ii — Integrals

Let $\vec{a}, \vec{b},\vec{c}$ be vectors. All of the following are defined EXCEPT:

Possible Answers:

$(\vec{a}\cdot \vec{b})+(\vec{a}\cdot \vec{c})$

$(\vec{a}\cdot \vec{b})\times\vec{c}$

$(\vec{a}\times \vec{b})\cdot(\vec{a}\times \vec{c})$

$(\vec{a}\times \vec{b})\cdot \vec{c}$

$(\vec{a}\times \vec{b})\times\vec{c}$

Correct answer:

$(\vec{a}\cdot \vec{b})\times\vec{c}$

Explanation:

The cross product of two vectors (represented by "x") requires two vectors and results in another vector. By contrast, the dot product (represented by " $\cdot$ ") between two vectors requires two vectors and results in a scalar, not a vector.

If we were to evaluate $(\vec{a}\cdot \vec{b})\times\vec{c}$ , we would first have to evaluate $\vec{a}\cdot\vec{b}$ , which would result in a scalar, because it is a dot product.

However, once we have a scalar value, we cannot calculate a cross product with another vector, because a cross product requires two vectors. For example, we cannot find the cross product between 4 and the vector <1, 2, 3>; the cross product is only defined for two vectors, not scalars.

The answer is $(\vec{a}\cdot \vec{b})\times\vec{c}$ .

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Example Question #1 :Understanding Vector Calculations

Find the magnitude of vector v=<3,-2> :

Possible Answers:

$\sqrt{11}$

$\sqrt{13}$

$\sqrt{7}$

$\sqrt{5}$

Correct answer:

$\sqrt{13}$

Explanation:

To solve for the magnitude of a vector, we use the following formula:

$\left \| v \right \|=\sqrt{x^2+y^2}$

v=<3,-2>

$\left \| v \right \|=\sqrt{(3)^2+(-2)^2}$

$\left \| v \right \|=\sqrt{9+}4$

$\left \| v \right \|=\sqrt{13}$

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Example Question #1 :Understanding Vector Calculations

Given vector u=<0,0> and v=<2,1> , solve for u+v .

Possible Answers:

<2,0>

<2,1>

<0,0>

<0,1>

Correct answer:

<2,1>

Explanation:

u=<0,0> v=<2,1>

To solve for u+v ,我们需要添加components in the vector and thecomponents together:

u+v=<0+2,0+1>

u+v=<2,1>

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Example Question #1 :Parametric, Polar, And Vector

Given vector u=<0,0> and v=<2,1> , solve for u-v .

Possible Answers:

<0,-1>

<2,0>

<2,1>

<-2,-1>

Correct answer:

<-2,-1>

Explanation:

u=<0,0> v=<2,1>

To solve for u-v , we need to subtract thecomponents in the vector and thecomponents together:

u-v= <0-2,0-1>

u-v= <-2,-1>

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High School Math : Parametric, Polar, and Vector

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 :Understanding Polar Coordinates

Example Question #2 :Calculus Ii — Integrals

Example Question #3 :Understanding Polar Coordinates

Example Question #1 :Understanding Polar Coordinates

Example Question #1 :Calculus Ii — Integrals

Example Question #1 :Parametric, Polar, And Vector

Example Question #11 :Calculus Ii — Integrals

Example Question #1 :Understanding Vector Calculations

Example Question #1 :Understanding Vector Calculations

Example Question #1 :Parametric, Polar, And Vector

All High School Math Resources

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