Math 233 Vectors! Version: 8-29-05
Know The Difference!
Scalar: a quantity represented by a real number
Vector: a quantity represented by both a direction and a magnitude
Displacement Vector: measures the displacement between two points
A=(2, 3) and B=(5,2), then AB 5 2, 2 3 3, 1 Always remember: Head – Tail or To – From
Magnitude: the magnitude of a vector a, b, c is: a 2 b2 c 2
Unit Vector: a vector whose magnitude is 1 (so it measures direction, or relative slope
Notation: b b 4,2 4i 2 ˆ where iˆ and ˆ are unit vectors in the positive x-
ˆ j j
and y- axis directions.
More Notation: unit vectors are frequently given “hats” (or carets) to denote their
unique property ( iˆ ).
Equality: Two vectors are equal if they have the same length and direction
1. Find the displacement from the point P1 to P2 for P 4,6
1 P2 7,10 :
2. Given the vectors u and v shown, sketch the vectors 2u , u v , and u v .
3. Two adjacent sides of a regular hexagon are given as the vectors u and v . Label the
remaining sides in terms of u and v (for example, u or u v ).
4. Given the vector v 2,4 .
a) What is the magnitude of v (noted as v or sometimes v )?
(Hint: draw a picture and then use the Pythagorean Theorem)
b) Find a unit vector in the direction of v .
(Hint: the magnitude of a unit vector is 1… what was the magnitude of v ?)
5. A vector has length 8 and makes a 30-degree angle with the x-axis. What are the values
of the iˆ and ˆ components?
6. The velocity of a truck is determined to be 9i 11 ˆ . If another truck is traveling twice as
fast in the opposite direction, what is the velocity vector of the second truck? What is
the speed of the second truck (note: speed is a scalar, velocity is a vector)?
7. Super Bonus Problem: An airplane is flying at an airspeed of 600 km/hr in a cross-wind
that is blowing from the northeast at a speed of 50 km/hr. In what direction should the
plane head to end up going due east? What will the groundspeed end up being (this is the
speed in the resultant, or due east, direction)?