Given c = (2.4,0.45) and d = (7,-4) find the direction of 5c+4d

The direction of 5c + 4d is approximately 19.0° clockwise with

the positive part of x-axis

Step-by-step explanation:

* Lets talk about the direction of a vector

- If the vector is (x, y), then

# Its magnitude is √ (x² + y²)

# Its direction is tan^-1 (y/x)

- The direction is the angle between the positive part of x-axis and

the vector

* Lets solve the problem

∵ c = (2.4, 0.45) and d = (7, - 4)

- To find 5c + 4d, multiply the two coordinates of c by 5 and the two

coordinates of d by 4

# Multiply c by 5

∴ 5c = [5 (2.4), 5 (0.45) ]

∴ 5c = (12, 2.25)

# Multiply d by 4

∴ 4d = [4 (7), 4 (-4) ]

∴ 4d = (28, - 16)

- Lets add 5c and 4d

∴ 5c + 4d = (12, 2.25) + (28, - 16)

∴ 5c + 4d = [ (12 + 28), (2.25 + - 16) ]

∴ 5c + 4d = (40, - 13.75)

- The vector is in the 4th quadrant because the x-coordinate is

positive and the y-coordinate is negative

* Find the direction of 5c + 4d

∵ The direction is tan^-1 (y/x)

∴ The direction is tan^-1 (-13.75/40) = - 18.97°

* The direction of 5c + 4d is approximately 19.0° clockwise with

the positive part of x-axis