Question

Vector A has magnitude of 8units and make an angle of 45° with positive x-axis vector B has the same magnitude of 8units the dirction alone the negative x-axis find

a, the magnitude and the dirction A+B

Answer 1

The magnitude and direction of vector A + B is approximately 6.28 units and -67.5° from the positive x-axis, respectively.

Step-by-step explanation:

The student is asking about vector addition and how to find the magnitude and direction of the resultant vector when two vectors are combined. In the given problem, vector A has a magnitude of 8 units and makes a 45° angle with the positive x-axis, and vector B also has a magnitude of 8 units but is directed along the negative x-axis.

To find vector A + B, we need to calculate the x and y components of each vector. For vector A:

• Ax = A * cos(θ) = 8 * cos(45°) = 8 * 0.7071 ≈ 5.66 units (x-component)
• Ay = A * sin(θ) = 8 * sin(45°) = 8 * 0.7071 ≈ 5.66 units (y-component)

Since vector B is along the negative x-axis, its y-component is 0:

• Bx = -8 units (x-component, negative because it's in the negative direction)
• By = 0 units (y-component)

To find the resultant vector R = A + B, we add the components of A and B:

• Rx = Ax + Bx = 5.66 - 8 ≈ -2.34 units
• Ry = Ay + By = 5.66 + 0 = 5.66 units

Now, we calculate the magnitude and direction of R:

Magnitude of R = √(Rx^2 + Ry^2)

= √((-2.34)^2 + (5.66)^2)

≈ √(39.39)

≈ 6.28 units

Direction of R = arctan(Ry/Rx) = arctan(5.66/-2.34)

≈ arctan(-2.42)

≈ -67.5° from the positive x-axis.