MCV4U Introduction to Vectors Practice Test (with Solutions)

Free Geometric Vectors Practice Test

8 questions (15 parts), exactly as a free account gets them in the first Geometric Vectors test on Martian Lab. Try each one on paper first, then open the answer.

1. Consider the given diagram then describe the direction of $\vec{u}$ using

   

   1. true bearing
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      $135^{\circ}$
   2. quadrant bearing
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      $[S\,45^{\circ}\,E]$
2. Describe how the vectors $3\vec{v}$ and $-3\vec{v}$ are related $\vec{v}$. Include diagrams in your description.

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   They are collinear or parallel. They have different magnitude. $\vec{v}$ and $3\vec{v}$ have the same direction. $\vec{v}$ and $-3\vec{v}$ have opposite directions.
3. Consider rectangle $ABCD$ as shown then fill in the blanks with the most appropriate words relating to vectors.

   

   1. Vectors $\rule{1cm}{0.4pt}$ and $\rule{1cm}{0.4pt}$ are equivalent vectors.
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      $\overrightarrow{AD}$ $\overrightarrow{BC}$
   2. $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are $\rule{1cm}{0.4pt}$ vectors.
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      opposite
   3. $\overrightarrow{AE}$ and $\overrightarrow{CA}$ are $\rule{1cm}{0.4pt}$ vectors.
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      collinear
4. In the given diagram, $WXYZ$ is a parallelogram, $C$ is the midpoint of $WZ$, and $|\overrightarrow{XA}| = |\overrightarrow{AB}| = |\overrightarrow{BY}|$. Express each vector below in terms of $\vec{a}$, $\vec{b}$, $\vec{m}$, and/or $\vec{n}$.

   

   1. $\overrightarrow{AB} = \rule{1cm}{0.4pt}$
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      $-\vec{m} - \vec{n}$
   2. $\overrightarrow{AZ} = \rule{1cm}{0.4pt}$
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      $-\vec{m} + \vec{b}$
   3. $\overrightarrow{WB} = \rule{1cm}{0.4pt}$
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      $\vec{b} - \vec{n}$
   4. $\overrightarrow{XY} = \rule{1cm}{0.4pt}$
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      $3(\vec{m} + \vec{n})$
   5. $\overrightarrow{YZ} = \rule{1cm}{0.4pt}$
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      $\vec{a}$
5. Given $\vec{v} = 12\,\text{m/s}$ at a bearing of $020^{\circ}$. What are the horizontal and vertical components, $\vec{v}_h$ and $\vec{v}_v$ respectively, of the vector?

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   $|\vec{v}_y| = 12 \times \sin 70^{\circ}$ $\approx 11.28\,\text{m/s}$ $|\vec{v}_x| = 12 \cos 70^{\circ}$ $\approx 4.10\,\text{m/s}$
6. Given $|\vec{a}| = 8$, $|\vec{b}| = 12$, and $|\vec{a} - \vec{b}| = 10$, determine the angle between the vectors $\vec{a}$ and $\vec{b}$ to the nearest degree. Include a diagram in your answer.

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   $|\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2 \times |\vec{a}|\,|\vec{b}|\cos\theta$ $100 = 64 + 144 - 2 \times 8 \times 12 \times \cos\theta$ $\cos\theta = \dfrac{108}{192},\; \theta \approx 55.8^{\circ}$ $\therefore$ Angle between the two vectors is about $55.8^{\circ}$.
7. If the angle between unit vectors $\hat{a}$ and $\hat{b}$ is $120^{\circ}$, what is $|8\hat{a} - 5\hat{b}|$? Include a diagram in your answer.

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   $|8\hat{a} - 5\hat{b}|^2 = 8^2 + 5^2 - 2 \times 8 \times 5 \times \cos 120^{\circ}$ $= 129$ $|8\hat{a} - 5\hat{b}| = \sqrt{129}$ $\approx 11.4$
8. Evan is towing his brother on a toboggan, using a rope which makes an angle of $20^{\circ}$ with the ground. If Evan is pulling with a force of $55\,\text{N}$, what horizontal force is he exerting on the toboggan? Include a diagram in your answer.

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   $|\vec{F}_x| = 55\cos 20^{\circ} \approx 51.7$ $\therefore$ The horizontal force is about $51.7\,\text{N}$.

This Unit on Martian Lab

The lessons start with geometric vectors, then apply them to force and to velocity.

1. Geometric Vectors
2. Applications of Vectors: Force
3. Applications of Vectors: Velocity

Then 4 full Geometric Vectors practice tests to check yourself.

Geometric Vectors FAQ

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