Adding Perpendicular Vectors
Activities and Notes
vector addition – combining vectors by placing the tail of the second vector to the head of
the first vector; the sum of the vectors show a third vector; the directions of the original
vectors do not change but are simply moved.
resultant – vector sum of two or more vectors; drawn from the tail of the first vector to
the head of the last vector.

Draw the vectors that represent a person walking 100 meters east, pausing, then
walking another 100 meters east. Draw another vector if the person went back 50
meters, or 50 meters west.
When drawing vectors to scale, a ruler can be used to accurately measure the magnitude
of the resultant vector when working in one-dimension.

Draw the vectors that represent a person walking 30 meters east, then 40 meters
north.
Drawing two-dimensional vectors graphically follows the same rules for one-dimensional
vectors. Place the tail of the second vector to the head of the first vector. Draw the
resultant vector by connecting the tail of the first vector to the head of the last vector.
The direction of the resultant vector must be included. The vector’s direction is show as
an angle, measured from the horizontal. If drawn to scale, a ruler is used to measure the
magnitude of the resultant vector and a protractor is used to measure the angle 
(direction) of the resultant vector.
horizontal – zero line; used as a point of reference for measuring vector angles.
The direction  of the resultant vector is found by placing a coordinate system with the
origin at the tail of the resultant vector. From the horizontal (x-axis), start in quadrant I
and move counterclockwise until the angle reaches the resultant vector. The angle made
is the direction  of the resultant vector.
It is not always convenient to measure the magnitude and direction of a vector. Using
trigonometry, the magnitude and direction of the resultant vector can be calculated.
Recall the right triangle with respect to the trigonometric functions sine, cosine, and
tangent: SOH – CAH – TOA.
sin  
opposite
hypotenuse
cos  
adjacent
hypotenuse
tan  
opposite
adjacent
To mathematically solve for any unknown angle  use the trigonometric functions for
sine, cosine, and tangent. To calculate an unknown side of the right triangle, use the
Pythagorean Theorem a2 + b2 = c2 where a and b are the legs of the triangle and c
represents the hypotenuse.
Determine the magnitude and direction of the resultant vector that represents a person
walking 30 meters east or 0, then 40 meters north or 90.
Using the Pythagorean Theorem the resultant vector has a magnitude of 50 meters.
a 2  b 2  c 2  c  a 2  b 2  (30m) 2  (40m) 2  900m 2  1600m 2  2500m 2  50m
The direction  is found using SOH – CAH – TOA. Since the opposite and adjacent sides
are already known, TOA is used.
tan  
opposite 40m

 1.33    tan 1 1.33  53.1
adjacent 30m
The resultant vector is 50 meters @ 53.1.
The resultant vector of a person walking shows the displacement of the person, not the
total distance traveled. For the above problem, the person walked a distance of 70
meters, but the displacement, or change in position, was only 50 meters.
Change the order the person walks to see if the displacement, distance walked, or
resultant vector changes. Determine the magnitude and direction of the resultant vector
that represents a person walking 40 meters north or 90, then 30 meters east or 0.
Using the Pythagorean Theorem the resultant vector has a magnitude of 50 meters.
a 2  b 2  c 2  c  a 2  b 2  (40m) 2  (30m) 2  1600m 2  900m 2  2500m 2  50m
The angle or direction of the resultant vector  is not simply the angle inside the right
triangle. The direction is always measured from the horizontal and counterclockwise to
the resultant vector. Instead, trigonometry can be used to find an angle  inside the right
triangle which can be used to calculate .
Since the opposite and adjacent sides are already known, TOA is used to find .
tan  
opposite 30m

 0.75    tan 1 0.75  36.9
adjacent 40m
Since  and  form a right angle of 90,  is found from 90 -  = 90 - 36.9 = 53.1.
The resultant vector is 50 meters @ 53.1.
The order you add the vectors does not change the magnitude or direction of the
resultant vector. The complexity of the problem is affected depending on the order the
vectors are added.
Summary Review – Vector Basics and Adding Perpendicular Vectors

A vector is used to graphically represent vector quantities. The arrow-tip is called
the head of the vector and the open end is called the tail of the vector.

Vectors are added by placing the tail of a vector to the head of the previous vector.

A resultant vector is the vector sum of two or more vectors. They are drawn from
the tail of the first vector to the head of the last vector.

The direction or angle of a vector is always measured from the horizontal.

Trigonometric functions and the Pythagorean Theorem are used to calculate the
magnitude and direction of resultant vectors.