Final answer:
To find the magnitude and direction of vector D, rearrange the given equation and substitute the values for vector B and vector C. Simplify the equation and isolate vector D. Calculate the magnitude of vector D using the formula and determine its direction using the arctan function.
Step-by-step explanation:
To determine the magnitude and direction of vector D, we can rearrange the equation 3B - C + D = 0 to solve for D. First, substitute the given values for vector B and vector C:
B = -i + 2j
C = 3i - 2j
Now, plug these values into the equation:
3(-i + 2j) - (3i - 2j) + D = 0
We can simplify the equation and isolate vector D:
-3i + 6j - 3i + 2j + D = 0
-6i + 8j + D = 0
D = 6i - 8j
The magnitude of vector D can be calculated using the formula:
|D| = sqrt((6)^2 + (-8)^2)
|D| = sqrt(36 + 64)
|D| = sqrt(100)
|D| = 10
The direction of vector D can be determined using the formula:
theta = arctan(Dy/Dx)
theta = arctan(-8/6)
theta = -53.13°
Therefore, the magnitude of vector D is 10 and its direction is -53.13°.-