Vectors & Scalars

Vector Addition

A scalar is a quantity that has magnitude only

A vector has both magnitude and direction

Example: 4 miles

Example: 4 miles north

Vectors are represented by arrows

The direction of the arrow represents the direction of the vector and the length of the arrow represents the magnitude

Vectors may be added by the Tip-To-Tail Method

Example: Four miles east plus three miles northThe additive is called the Resultant, and can be calculated using the Pythagorean Theorem.

Try these:

• 5 m/s E and 10 m/s N

• 5 m/s E and 10 m/s W

• 5 m/s N and 10 m/s E

Parallel Transport of Vectors

Vectors have a property known as parallel transport. The arrow representing a vector can be moved from one point to another, and as long as the length of the arrow and the direction of the arrow are unchanged, it is still the same vector.

Adding Three Vectors by the Tip-To-Tail Method

A

B C

A+B+C

The order of addition does not matter!

A+B+C = B+C+A

Vector Addition is COMMUTATIVE!

The sum of the vectors is the vector that goes from the tail of the first vector to the tip of the last vector

Place the vectors tip-to-tail

Add its negative.

Get the negative of a vector by reversing its direction.

A

-A

B

B - A

Parallel transport the vectors to form a parallelogram

Form the resultant vector: B + (-A) = B - A

To subtract a vector:

Another way:

• Law of SinessinA = sinB = sinC

a b c

There is also the Law of Cosines, but most students find the Law of Sines easier to work with!

Component Vectors

AB

AxBx

AyBy

Any vector can be split into component vectors.

Adding the component vectors gives the same result as adding the original vectors.

CCy

Cx

Cx = Ax + Bx

Cy = Ay + By

The vector C has component vectors Cx and Cy.

Components vs. Component Vectors

The x-component of a vector is (plus or minus) the length of the component vector Ax. The component is negative if the component vector is in the negative x-direction.

(On the previous slide, we get the length of Cx by subtracting the length of Bx from the length of Ax. We can still write Cx = Ax + Bx, though, because Bx is negative.)

Similarly, the y-component of a vector is (plus or minus) the length of the component vector Ay. The component is negative if the component vector is in the negative y-direction.

NOTE: Usually the component vectors are written in bold and the components are not.

Calculating Vector Components

Ay

Ax

A

sin = opposite / hypotenuse =Ay

A

cos adjacent / hypotenuse= =Ax

A

Due to the way sine and cosine are defined for angles greater than 90º, if the angle is measured relative to the positive x-axis, the components will always have the correct sign when these equations are used.

What if you don’t want to measure the angle relative to the positive x-axis?

Use the following technique:

A

Ax

Ay

Ay has length A cos because Ay is adjacent to . Ay is positive because it is in the positive y-direction

Ax has length A sin because Ax is opposite . Ax is negative because it is in the negative x-direction

Enclose the known angle with the component vectors.

What if I know the other angle?

Ay has length A sin because Ay is opposite to . Ay is positive because it is in the positive y-direction

Ax has length A cos because Ax is adjacent to . Ax is negative because it is in the negative x-direction

We may specify a (two-dimensional) vector in one of two ways:

We give its components, Ax and Ay.Example: Ax = 4 miles east and Ay = 3 miles northWe call this Rectangular Form

We give the magnitude and direction of the vector, A and In this example A = 5 miles and = 37ºWe call this Polar Form

The two forms are completely equivalent!

We already know how to calculate Ax and Ay from A and .What about the reverse?

4

337º

5

A

Ay

Ax

Pythagorian Theorem:A

2Ax

2 + Ay2=

tan = opposite / adjacent = Ay / Ax( )

-1

Calculating Magnitude and Direction of a Vector from its Components

CAUTION!

There is a possible ambiguity with the inverse tangent!

Because the tangent function repeats every 180º[i.e., tan( + 180º) = tan ], the inverse tangent function on your calculator will return results between - 90º and + 90º!

SOLUTION: If the vector is in the second or third quadrant (when Ax is negative) ADD 180º TO THE RESULT GIVEN BY YOUR CALCULATOR