##### Notes

This printable supports Common Core Mathematics Standard HSN-VM.B.4, HSN-VM.B.4b

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## Vector Addition Given Magnitude and Direction

Instructions: Unless otherwise stated, the direction angle is the angle between the positive x-axis and the vector, starting from the positive x-axis and going in a counterclockwise direction.

1.
Vector $vec a$ has a magnitude of 5 and its direction is $30 deg$ East of North. Vector $vec b$ has a magnitude of 3 and its direction is $30 deg$ East of North. What is the resulting magnitude of $vec a + vec b$ ?
1. 8
2. 2
3. 5.8
4. 4
2.
Vector $vec a$ has a magnitude of 4 and its direction is $10 deg$ West of North. Vector $vec b$ has a magnitude of 6 and its direction is $80 deg$ South of East. What is the resulting magnitude of $vec a + vec b$ ?
1. 1.92
2. 0.35
3. 2.00
4. 2.03
3.
Vector $vec a$ has a magnitude of 4 and its direction is $10 deg$ West of North. Vector $vec b$ has a magnitude of 6 and its direction is $80 deg$ South of East. What is the resulting direction of $vec a + vec b$ ?
1. $10 deg$ East of North
2. $70 deg$ North of West
3. $80 deg$ East of South
4. $10 deg$ East of South
4.
What is the sum of a vector with magnitude 6 and direction 35° and a vector with magnitude 12 and direction 190°?
1. Magnitude 7.0, direction 168.9°.
2. Magnitude 17.6, direction 133.9°.
3. Magnitude 10.7, direction 135.5°.
4. Magnitude 15.7, direction 170.5°.
5.
If $vec{a}$ has magnitude 21 and direction 55°, and $vec{b}$ has magnitude 15 and direction 320°, what is the magnitude and direction of $vec{a} + vec{b} ?$
1. Magnitude 22.9, direction 57.8°.
2. Magnitude 25.3, direction 37.2°.
3. Magnitude 26.8, direction 357.9°.
4. Magnitude 24.7, direction 17.8°.
6.
If $vec{v}$ has magnitude of 8 and direction 200°, and $vec{w}$ has magnitude of 10 and direction 265°, what is the magnitude and direction of $vec{v} + vec{w} ?$
1. Magnitude 9.8, direction 36.5°.
2. Magnitude 3.7, direction 163.5°.
3. Magnitude 15.2, direction 236.5°.
4. Magnitude 12.2, direction 213.9°.
7.
If $||vec{n}|| = 2||vec{m}||$, the direction of $vec{n}$ is 40°, and the direction of $vec{m}$ is 160°, in which quadrant does the sum of these vectors lie, and why?
1. Quadrant I, since the resulting direction of the sum of the vectors is 70°.
2. Quadrant I, since the resulting direction of the sum of the vectors is 30°.
3. Quadrant II, since $vec{m}$ is much farther from the positive x-axis than $vec{n}$ (even though $||vec{m}|| < ||vec{n}||$ ).
4. Along the positive y-axis (so not in any quadrant), since the direction of the sum of the vectors is 90°.
8.
For $vec{a}$, its magnitude is 10 and its direction is 45°. If the direction of $vec{b}$ is 135°, for what magnitudes of $vec{b}$ will the vector sum $vec{a} + vec{b}$ be in the first quadrant?
1. True for all magnitudes of $vec{b}$.
2. $||vec{b}|| > 10$
3. $||vec{b}|| < 10$
4. $||vec{b}|| < 5$
9.
Two boats are pulling a very large, derelict boat behind them. One boat is exerting a force of $1.1 xx 10^6$ N on the derelict boat in the direction S10°W. The second boat is pulling with a force of $1.3 xx 10^6$ N in the direction S5°E. What is the resulting force acting on the derelict boat and in what direction? Round answers to the nearest whole unit.
1. The resulting force is $1.4 xx 10^6$ N, in the direction S7°E.
2. The resulting force is $2.4 xx 10^6$ N, in the direction S2°W.
3. The resulting force is $3.7 xx 10^5$ N, in the direction S8°W.
4. The resulting force is $1.6 xx 10^6$ N, in the direction S2°E.
10.
Jim and Sarah are trying to push a heavy box across the floor. The box is at the back of the room, and they are pulling it towards the door which is directly in front of them. They have attached ropes to the center of the front of the box, but neither of them is able to pull it completely straight forwards. Jim is pulling with 600 N of force at an angle of 15° left of the door. Sarah is pulling with 650 N of force at an angle of 20° right of the door. How much force is actually being exerted on the box, and in what direction will it move? Round answers to the nearest whole unit.
1. The box is being pulled with a force of 379 N in a direction of 17° right of the door.
2. The box is being pulled with a force of 1192 N in a direction of 3° right of the door.
3. The box is being pulled with a force of 1049 N in a direction of 18° left of the door.
4. The box is being pulled with a force of 1250 N 5° right of the door.
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