Given the vectors = (19 cm, -7 cm) in green and = (-23 cm, 9 cm) in blue shown on the grid below, calculate the magnitude of the vectors described below and select the correct sketch for each from the choices below. The tail-to-head method is used to represent the operations performed on the vectors, and the scale in the figures is centimeters.
To calculate the magnitude of a vector, you can use the Pythagorean theorem. The magnitude of a vector is the square root of the sum of the squares of its components.
For example, to find the magnitude of vector A with components (19 cm, -7 cm), you can use the formula:
|A| = sqrt((19 cm)^2 + (-7 cm)^2)
Simplifying this equation, we get:
|A| = sqrt(361 cm^2 + 49 cm^2)
= sqrt(410 cm^2)
Thus, the magnitude of vector A is approximately sqrt(410) cm.
Similarly, you can find the magnitude of vector B with components (-23 cm, 9 cm) using the same method.
|B| = sqrt((-23 cm)^2 + (9 cm)^2)
Simplifying, we get:
|B| = sqrt(529 cm^2 + 81 cm^2)
= sqrt(610 cm^2)
Thus, the magnitude of vector B is approximately sqrt(610) cm.
Now, to select the correct sketch for each vector, compare the calculated magnitudes to the given sketches and determine which one matches.