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Trigonometry : Vectors

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Example Questions

Example Question #1 : Vectors

Determine the magnitude of vector A.

Possible Answers:

$|\vec{A}| = 25$

$|\vec{A}| = 12$

$|\vec{A}| = 16$

$|\vec{A}| = 5$

Correct answer:

$|\vec{A}| = 5$

Explanation:

We can use the pythagorean theorem to solve this problem. Using $\vec{A}$ as our hypotenuse, we can drop a vertical vector perpendicular to the x-axis. We will call this $\vec{A_y}$ and it is 4 units in length. We can also extend a vector from the origin that connects to $\vec{A_y}$ . We will call this $\vec{A_x}$ and it is 3 units in length.

Using the pythagorean theorem:

$|\vec{A}|^2 = \vec{A_x}^2 + \vec{A_y}^2$

$|\vec{A}| = \sqrt{\vec{A_x}^2 + \vec{A_y}^2}$

$|\vec{A}| = \sqrt{3^2 + 4^2}$

$|\vec{A}| = \sqrt{9 + 16}$

$|\vec{A}| = \sqrt{25}$

$|\vec{A}| = 5$

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Example Question #1 : Vectors

Which of the following is the correct term for the sum of two vectors?

Possible Answers:

Resultant

Magnitude

Sum

Velocity

Correct answer:

Resultant

Explanation:

When summing two vectors, you have both an x and y component and you sum these separately leaving you with a coordinate as your answer. This coordinate is called a resultant.

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Example Question #1 : Vectors

Determine the resultant of $\vec{A} = (2,5)$ and $\vec{B} = (1,9)$ .

Possible Answers:

$\vec{A} + \vec{B} = (3, 14)$

$\vec{A} + \vec{B} = (1, -4)$

$\vec{A} + \vec{B} = 17$

$\vec{A} + \vec{B} = (14, 3)$

Correct answer:

$\vec{A} + \vec{B} = (3, 14)$

Explanation:

When determining the resultant of two vectors, you are finding the sum of two vectors. To do this you must add the x component and the y component separately.

$\vec{A}+\vec{B} = (2,5) + (1,9)$

$\vec{A}+\vec{B} = (2 + 1, 5 + 9)$

$\vec{A} + \vec{B} = (3, 14)$

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Example Question #2 : Vectors

Consider the following graphs where $\vec{A}$ begins at the origin and ends at (-3, 5) and $\vec{B} = (4, 1)$ . Which of the following depicts the correct resultant of these two vectors.

Screen shot 2020 08 27 at 1.29.58 pm

Possible Answers:

Screen shot 2020 08 27 at 1.33.16 pm

Screen shot 2020 08 27 at 1.32.49 pm

Screen shot 2020 08 27 at 1.32.16 pm

Screen shot 2020 08 27 at 1.31.51 pm

Correct answer:

Screen shot 2020 08 27 at 1.33.16 pm

Explanation:

To find the resultant we must sum the two vectors:

$\vec{A} + \vec{B} = (-3, 5) + (4, 1)$

$\vec{A} + \vec{B} = (-3 + 4, 5 + 1)$

$\vec{A} + \vec{B} = (1, 6)$

Now we must graph the resultant

Screen shot 2020 08 27 at 1.31.14 pm

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Example Question #3 : Vectors

How many degrees above the x-axis is $\vec{A}$ ?

Screen shot 2020 08 27 at 2.25.43 pm

Possible Answers:

43.7^o

23.2^o

66.8^o

113.2^o

Correct answer:

23.2^o

Explanation:

First, we must understand what we are solving for. We are solving for the angle that is formed by $\vec{A}$ and the x-axis. To do this, we can extend a vector from the origin which stops directly under the end of $\vec{A}$ . We will call this new vector $\vec{A_x}$ and it will be 7 units long. We will also extend a vector upwards that is perpendicular to the x-axis. We will call this $\vec{A_y}$ and it will be 3 units long.

Screen shot 2020 08 27 at 2.26.24 pm

Now we can use the relationship that $tan(\theta) = \frac{opposite}{adjacent}$ where $\vec{A_x}$ is the adjacent side and $\vec{A_y}$ is the opposite side.

$tan(\theta) = \frac{\vec{A_y}}{\vec{A_x}}$

$tan(\theta) = \frac{3}{7}$

$\theta = tan_{-1}(\frac{3}{7})$

$\theta = 23.2$

And so $\vec{A}$ is 23.2 degrees above the x-axis.

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Example Question #1 : Vectors

Find the difference of the two vectors, $\vec{A}$ which ends at (-2, 4) and $\vec{B}$ ending at (7, -6) .

Possible Answers:

(-9, 10)

Correct answer:

(-9, 10)

Explanation:

When finding the difference of two vectors, you must subtract the x and y components separately.

$\vec{A} - \vec{B} = (-2, 4) - (7, -6)$

$\vec{A} - \vec{B} = (-2-7, 4 - (-6))$

$\vec{A} - \vec{B} = (-9, 10)$

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Example Question #5 : Vectors

Which of the following is the correct depiction of the difference of vectors A and B?

Screen shot 2020 08 27 at 1.42.05 pm

Possible Answers:

Screen shot 2020 08 27 at 1.43.44 pm

Screen shot 2020 08 27 at 1.43.37 pm

Screen shot 2020 08 27 at 1.43.01 pm

Screen shot 2020 08 27 at 1.43.50 pm

Correct answer:

Screen shot 2020 08 27 at 1.43.01 pm

Explanation:

To find the difference of two vectors we must consider the x and y components separately.

$\vec{A} - \vec{B} = (5, 8) - (-3, 10)$

$\vec{A} - \vec{B} = (5 - (-3), 8 - 10)$

$\vec{A} = \vec{B} = (8, -2)$

And then we must correctly graph this vector

Screen shot 2020 08 27 at 1.43.01 pm

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Example Question #6 : Vectors

True or False: The magnitude of a vector is the length of the vector.

Possible Answers:

False

True

Correct answer:

True

Explanation:

When finding the magnitude of the vector, you use either the Pythagorean Theorem by forming a right triangle with the vector in question or you can use the distance formula. This is much more clear considering the distance vector that the magnitude of the vector is in fact the length of the vector.

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Trigonometry : Vectors

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Vectors

Example Question #1 : Vectors

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