To find the vector difference of a and b, we will subtract vector b from vector a:
a – b = (400i + 300j) – (500i + 200j)
First, we can break this down into its components:
- i-components: 400 – 500 = -100
- j-components: 300 – 200 = 100
So the resulting vector difference is:
a – b = -100i + 100j
Next, to find the magnitude of this vector, we can use the formula:
|v| = √(x² + y²)
Where x and y are the components of the vector. In our case:
|v| = √((-100)² + (100)²)
This simplifies to:
|v| = √(10000 + 10000) = √20000 = 141.42
Now, we will find the direction of the vector using the tangent function. The angle θ can be found with:
tan(θ) = y/x = 100 / (-100)
This gives us:
tan(θ) = -1
Now we can find the angle:
θ = arctan(-1)
This corresponds to an angle of -45 degrees. Since the vector is in the second quadrant (negative i and positive j), we add 180 degrees to find the angle from the positive x-axis:
θ = -45 + 180 = 135 degrees
In conclusion, the magnitude of the vector difference a – b is approximately 141.42 and the direction is 135 degrees from the positive x-axis.