Practice Direction in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The orientation of movement or facing in space, independent of speed or distance—where something points.

North, south, east, west—or the way an arrow points, regardless of how long the arrow is.

Showing a random 20 of 50 problems.

Example 1

easy
The unit vector in the direction of ⟨3,0⟩\langle 3, 0 \rangle is ____.

Example 2

hard
A ship sails N30°EN30°E (30° east of north). Express this direction as a unit vector in the standard (x,y)(x, y) coordinate system where x is east and y is north.

Example 3

medium
Convert the vector ⟨3,4⟩\langle 3, 4 \rangle into a magnitude and a direction angle from the positive xx-axis.

Example 4

easy
Turning from north to east is a turn of how many degrees?

Example 5

medium
Convert a bearing of N60∘EN60^\circ E to a standard angle from the positive xx-axis (east) measured counterclockwise.

Example 6

easy
What compass direction has bearing 180∘180^\circ?

Example 7

medium
A bearing is measured clockwise from north. What bearing is due west?

Example 8

easy
Two arrows have the same direction but different lengths. Are they the same vector?

Example 9

easy
Opposite direction of east is ____.

Example 10

challenge
Why does specifying a direction in 3D space require two angles, while in 2D it needs only one?

Example 11

medium
A vector points at a bearing of 45∘45^\circ (northeast). Its components are equal. If its magnitude is 2\sqrt{2}, find the components.

Example 12

challenge
In 3D, a direction can be parameterized by two angles (e.g., azimuth and elevation). Explain why a unit vector ⟨x,y,z⟩\langle x, y, z \rangle has only 2 degrees of freedom even though it has 3 components.

Example 13

medium
A unit vector represents pure direction. Why is its magnitude always 1?

Example 14

easy
A vector ⟨5,0⟩\langle 5, 0 \rangle points in which direction?

Example 15

easy
Do two objects moving the same distance always move in the same direction?

Example 16

medium
A boat heads east at 5 m/s; the current pushes it north at 3 m/s. Find the direction of the boat's resulting velocity.

Example 17

easy
Do the vectors (2,4)(2, 4) and (1,2)(1, 2) point in the same direction? Explain.

Example 18

hard
Why does a unit vector in 2D have just one degree of freedom while a vector in 2D has two?

Example 19

medium
Two unit vectors that point in opposite directions sum to ____.

Example 20

easy
What is the opposite direction of 'north'?
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