An Introduction to Vectors

We have discussed briefly the terms scalar and vector. A scalar quantity is one that has magnitude only. Speed, distance, temperature, and time are examples of scalar quantities. Vectors have both magnitude and direction. Displacement, velocity, acceleration, force, and momentum are examples of quantities in physics that need both a number to indicate their size and a direction to indicate which way they are going.

The symbol for the vector itself is just an arrow, where the length of the arrow represents a relative magnitude and the direction the arrow is pointing represents the direction of the quantity.

The addition of two or more vectors yields a vector called the resultant. There are two ways that we can obtain a resultant vector: graphically and algebraically. Let's start graphically, as this can give us a good model of the motion of objects. To add vectors, simply place the vectors head to tail and the resulting vector is a line drawn from the beginning of the first vector to the end of the last vector. The direction and magnitude are given by the resultant direction and length. For example, a runner jogs 3 km North, then 6 km East. What is the displacement of the runner from the origin? In other words, how far from where he started is the runner?

 

The red arrow represents the resultant vector and its relative length and direction give the jogger's displacement from the origin. With a ruler and a protractor, we can determine what that is as long as we have drawn our first vectors to scale. There are some guidelines for adding vectors geometrically:

  • Vectors can be added in any order. As long as you follow the head to tail rule, you will end up with the same resultant. Using the above as an example:

This is true no matter how many vectors you add together or what order you place them in.

  • To subtract a vector, you add the negative, or opposite of a vector. The negative of a vector is simply that vector with the same magnitude but pointing in the opposite direction.
  • Multiplying a vector by a scalar results in a vector. The direction remains the same, but the magnitude changes according to the scalar quantity. For example, a car's velocity is 20 m/s east. If it doubles its velocity, it travels 40 m/s but it is still traveling east.

Algebraically, we can add any vectors by superimposing them upon an x and y axis and then using trigonometric functions (sine, cosine, and tangent) along with the Pythagorean Theorem to solve.

First, lets look at the trig functions for a right triangle:

If we define an angle q, then:

The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, or

 H2= O2 + A2.

This allows us to set up our method of solving vectors. Adding two vectors that are perpendicular is easy - we  ust add them as an adjacent and an opposite side of a right triangle and solve for the hypotenuse and the angle. For example, in our jogger's run at the top of the page, the magnitude of his displacement can be found:

and his direction can be found using the tangent function (actually, the arctangent) as

To be more accurate, we would draw a picture to show the angle, or say "26.6o north of east."

When two or more vectors are not perpendicular, we break each individual vector into an x and y component, add all of the x and y components, then solve the resulting right triangle for the hypotenuse. x axis values are given by the formula

x = h cos q,

and y values are given by the formula

y = h sin q,

where h is the length of the original vector and q is the angle of the vector.

For example: Directions are often given in terms of the compass, where North is 000o, East is 090o, South is 180o, and West is 270o. A pilot flying an airplane at 350 km/hr on a heading of 030o encounters a wind blowing towards the southeast (135o on a compass) at 50 km/hr. What is the resultant velocity (magnitude and direction) of the plane?

Steps:

  • Draw the picture so you have an idea of where the vectors are.
  • Break each vector into x and y components and add the components as shown in the
         table below (Hint - reference your angles from the positive x axis):

Vector

X

Y

350 km/hr at 030o (60o above x axis)

350 cos 60 = 150 km/hr

350 sin 60 = 303 km/hr

50 km/hr at 135o (45o below x axis)

50 cos -45 = 35km/hr

50 sin -45 =-35 km/hr

Totals

150+35=185km/hr

303-35=268 km/hr

Now, solve for the resultant vector using the tangent and the Pythagorean Theorem. The resulting magnitude is 325 km/hr. The angle is 55o above the x axis, or a heading of 035o. Looking at our vectors, this appears to make sense.

Often it is helpful to see what someone else has to say on the topic. Try http://www.physicsclassroom.com/Class/vectors/ for more information. Select one of the lessons on vectors to find additonal explanations.