Consider the two vectors A with arrow = 3 i − j and B with arrow = − i − 6 j.
Calculate the directions of A with arrow + B with arrow and A with arrow − B with arrow.
A with arrow + B with arrow ° (counterclockwise from the +x axis)
A with arrow − B with arrow ° (counterclockwise from the +x axis)
A+B = 2i-7j
tanθ = -7/2 in QIV
A-B = 4i+5j
tanθ = 5/4 in QI
To calculate the directions of A with arrow + B with arrow and A with arrow − B with arrow, we need to find the angles that these vectors make with the positive x-axis.
First, let's calculate the magnitudes of vectors A with arrow and B with arrow:
Magnitude of A with arrow (|A|):
|A| = √(A_x^2 + A_y^2)
|A| = √((3)^2 + (-1)^2)
|A| = √(9 + 1)
|A| = √10
Magnitude of B with arrow (|B|):
|B| = √(B_x^2 + B_y^2)
|B| = √((-1)^2 + (-6)^2)
|B| = √(1 + 36)
|B| = √37
Now let's calculate the angles:
Direction of A with arrow + B with arrow:
θ1 = tan^(-1)(B_y/A_x)
θ1 = tan^(-1)(-6/3)
θ1 = -63.43°
Direction of A with arrow − B with arrow:
θ2 = tan^(-1)((A_y - B_y)/(A_x - B_x))
θ2 = tan^(-1)((-1 - (-6))/(3 - (-1)))
θ2 = tan^(-1)(5/4)
θ2 = 51.34°
Therefore, the directions of A with arrow + B with arrow is -63.43° counterclockwise from the +x-axis, and the direction of A with arrow − B with arrow is 51.34° counterclockwise from the +x-axis.
To calculate the direction of the vector sum A with arrow + B with arrow, follow these steps:
1. Add the corresponding components of vector A with arrow and vector B with arrow.
A with arrow + B with arrow = (3i - j) + (-i - 6j)
= (3i - i) + (-j - 6j)
= 2i - 7j
2. Use the tangent function to find the angle theta (°) between the resulting vector and the positive x-axis.
theta = atan2(-7, 2)
3. Convert the obtained angle from radians to degrees.
theta_degrees = theta * (180/π)
4. The direction of A with arrow + B with arrow is counterclockwise from the positive x-axis and is equal to theta_degrees.
Similarly, to calculate the direction of the vector difference A with arrow − B with arrow, follow these steps:
1. Subtract the corresponding components of vector B with arrow from vector A with arrow.
A with arrow − B with arrow = (3i - j) - (-i - 6j)
= (3i - j) + (i + 6j)
= 4i - 5j
2. Use the tangent function to find the angle theta (°) between the resulting vector and the positive x-axis.
theta = atan2(-5, 4)
3. Convert the obtained angle from radians to degrees.
theta_degrees = theta * (180/π)
4. The direction of A with arrow − B with arrow is counterclockwise from the positive x-axis and is equal to theta_degrees.