The magnitude ∥v∥ is a vector's length; the direction is the angle it makes with a reference axis.
Magnitude is how long the arrow is—like measuring the length of a stick. Direction is which way it points. A unit vector is a 'pure direction' with length 1, like a compass needle. To get the unit vector, shrink or stretch the vector until its length is exactly 1 while keeping it pointed the same way.
Showing a random 20 of 50 problems.
Example 1
medium
Find the unit vector in the direction of ⟨3,4⟩.
Example 2
medium
Find the distance between points (1,2) and (4,6) using a vector.
Example 3
hard
Two forces of magnitude 10 N act at 0° and 120°. Find the magnitude of the resultant.Find the magnitude of the resultant of two 10N forces at 0° and 120°
Example 4
medium
A plane flies at 300 km/h in direction 30° north of east. Write its velocity in component form.Write the plane's velocity in component form
Example 5
challenge
Vectors u and v are perpendicular, ∥u∥=7, ∥v∥=24. Find ∥u+v∥.
Example 6
challenge
A vector has magnitude 13 and x-component 5 (first quadrant). Find its y-component.
Example 7
medium
Find the magnitude of ⟨−3,−4⟩.
Example 8
hard
Show that ∥u+v∥≤∥u∥+∥v∥ for u=⟨3,0⟩ and v=⟨0,4⟩.Illustrate ||u+v|| ≤ ||u||+||v|| for u=⟨3,0⟩ and v=⟨0,4⟩
Example 9
easy
What is the magnitude of ⟨0,0,0⟩?
Example 10
easy
Find the magnitude of ⟨−5,0⟩.
Example 11
challenge
Find the direction angle (from positive x-axis, 0°≤θ<360°) of v=⟨3,−4⟩.Find the direction angle (0°≤θ<360°) of v
Example 12
challenge
Find a unit vector perpendicular to ⟨3,4⟩.
Example 13
medium
Is ⟨0.6,0.8⟩ a unit vector?
Example 14
easy
Find the magnitude of ⟨0,7⟩.
Example 15
easy
Find ∥⟨8,15⟩∥.Find ||⟨8, 15⟩||
Example 16
hard
Find the direction angle of ⟨−2,−23⟩ measured from the positive x-axis.
Example 17
medium
Find the direction angle of ⟨−1,1⟩.Find the direction angle of ⟨-1, 1⟩
Example 18
easy
Find the magnitude of ⟨1,1⟩.
Example 19
medium
Write v in component form if ∥v∥=10 and its direction angle is 60°.Write v in component form (|v|=10, θ=60°)