Question

Vectors practice: Sketch, then calculate the magnitude and direction of the resultant for each of the following pairs of vectors.

a) Vector A = 3i + 4j, Vector B = -2i - 6j
b) Vector A = 2i - 5j, Vector B = 7i + 3j
c) Vector A = -i + 2j, Vector B = 4i - 3j
d) Vector A = 6i + 8j, Vector B = -4i - 12j

Answer 1

To find the magnitude and direction of the resultant for each pair of vectors, sketch the vectors, add them head-to-tail, and use trigonometry to determine the magnitude and direction of the resultant.

Step-by-step explanation:

In order to find the magnitude and direction of the resultant for each pair of vectors, we can use the graphical method of vector addition. First, sketch each vector on a graph. Then, use the head-to-tail method to add the vectors and draw the resultant vector from the tail of the first vector to the head of the last vector.

For example, in case (a), vector A = 3i + 4j and vector B = -2i - 6j. Sketching these vectors on a graph and adding them head-to-tail, we find that the resultant vector is R = i - 2j. The magnitude of R can be calculated using the Pythagorean theorem: |R| = sqrt((1)^2 + (-2)^2) = sqrt(5).

The direction of R can be found using trigonometry: tan(theta) = -2/1; theta = arctan(-2/1) = -63.4 degrees. Therefore, the magnitude of the resultant is sqrt(5) and the direction is -63.4 degrees.

Using the same method, we can find the magnitude and direction of the resultant for each pair of vectors (b), (c), and (d). The magnitudes and directions for these cases are as follows: (b) |R| = sqrt(74), theta = 13.3 degrees; (c) |R| = 9.2, theta = -20.5 degrees; (d) |R| = 16, theta = 150 degrees.