When 2 Vectors are at Right Angles to each otherUse the Pythagorean theorem to find the magnitude of the resultant vector. (a number and an angle)

R2 = Δx2 + Δy2

3.2 Vector Operations (Mathematical Addition of Vectors)

The Tangent Function Use the tangent function to find the direction (angle) of

the resultant vector.

The inverse of the tangent function indicates the angle theta (ϴ)

ϴ = tan -1 (opp) adj

tangent of angle = opposite leg

adjacent leg

“SOH” = sin ϴ = opp/hyp “CAH” = cos ϴ = adj / hyp“TOA” = tan ϴ = opp / adj

Sample Problem 3A: An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramid’s height is 136 m and its width is 2.30 x 102 m (230 m). What is the magnitude and the direction of the displacement of the archaeologist after she has climbed from the bottom of the pyramid to the top?Given: ∆ x = 115 m (half of 230)∆ y = 136 m

R = √(115 m)2 + (136 m)2

R = 178 m (magnitude)

R2 = Δx2 + Δy2

tan θ = Δy / Δx θ = tan-1 (136 m / 115 m)θ = 49.8° (direction)

When Vectors are not Perpendicular to each otherYou must Resolve Vectors into 2 Components X component parallel to x axis Y component parallel to y axis Can either positive or negative values

A truck carrying a film crew must be driven at the correct velocity to enable the crew to film the underside of the plane. The plane flies at 95 km/h at an angle of 20° relative to the ground. Find the truck’s velocity.

The hypotenuse (vplane) is the resultant vector that describes the airplane’s total velocity.

The adjacent leg represents the x component (vx), which describes the airplane’s horizontal speed.

The opposite leg represents the y component (vy), which describes the airplane’s vertical speed.

We are looking for the adjacent leg Vxcos θ = adjacent / hypotenuse

cos θ = vx / vplane

Rearrange and solve for Vx = velocity in x direction (truck’s velocity)

vx = (cos θ )(vplane)

vx = (cos 20°)(95 km/h)

vx = 90 km/h

Your Turn

Find the components of the velocity of a helicopter traveling 95 km/h at an angle of 35° to the ground.

How fast must 007 (Bond, James Bond) travel to stay beneath Dr. No’s airplane that is moving at 105 km/h at an angle of 25° to the ground?

A truck drives uphill with a 15° incline. If the truck has a constant speed of 22 m/s, what are the vertical and horizontal components of the truck’s velocity.

Adding Vectors at Angles (not perpendicular)

• When vectors to be added are not perpendicular, they do not form sides of a right triangle.

Look at the geometry for the situation

• Notice, the length of the horizontal component of the resultant vector is equal to the sum of the lengths of the horizontal components of the vectors that are being added together.

• This is also true for the vertical component.

Rx1

Ry1

Rx2

Ry2

Rx1 + Rx2

Ry1 + Ry2

Resultant Vector

Steps

1. Resolve each vector that is being added (addends) into components.

2. Add all the horizontal components together and all the vertical components together.

3. Use the Pythagorean Theorem and trig ratios to find the resultant vector.

Example:

• Add these 2 vectors together: 10 m/s at 0º and 12 m/s at 25º

Find the resultant vector, R at θ

10 m/s

12 m/s

25º

R

θ

Example:

Find components of each vector

10 m/s

12 m/s25º

R

θ

Vector 1

x y

10 cos(0) 10 sin(0)

Vector 2 12 cos(25) 12 sin(25)

Example for Case #3

Add horizontal and vertical components

10 m/s

12 m/s25º

R

θ

Vector 1

x y

10 cos(0) 10 sin(0)

Vector 2 12 cos(25) 12 sin(25)

21 5.1

Example:Find the magnitude and angle of the

resultant vector using the Pythagorean Theorem

R

θ

R2 = 212 + 5.12

R = 21.6 m/s

θ = tan-1 (5.1/21)

θ = 14º

R = 21.6 m/s at 14º

21

5.1