Question

Two position vectors lie in a plane. The first, vector ra, points at an angle of 20° below the positive x-axis and has a magnitude of 49.0 m. The second, vector rb, points at an angle of 45.0° above the positive x-axis and has a magnitude of 75 m.What are the properties of vectors ra and rb, including their magnitudes and angles with respect to the positive x-axis?

Answer 1

Vector ra has a magnitude of 49.0 m and points 20° below the positive x-axis, while vector rb has a magnitude of 75 m and points 45.0° above the positive x-axis. The properties of these vectors include their magnitudes and directions, and their components can be calculated using trigonometric functions with their respective angles.

Step-by-step explanation:

The student is asking about the properties of two position vectors that lie in a plane with respect to the positive x-axis. Vector ra points 20° below the positive x-axis and has a magnitude of 49.0 m. The vector rb points 45.0° above the positive x-axis and has a magnitude of 75 m.

To analyze these vectors, we use the components along the x- and y-axes. Components are found using trigonometric functions: for vector ra, since it is below the x-axis, its y-component will be negative. On the other hand, vector rb will have positive components in both the x and y directions because it is above the x-axis.

The vector's x-component can be calculated using the cosine of the angle (cosθ), and the y-component can be found using the sine of the angle (sinθ). For example, if ra has a magnitude of 49.0 m and it makes a 20° angle below the x-axis, its x-component would be 49.0 m * cos(20°) and its y-component would be -49.0 m * sin(20°), as the y-component is negative.

Answer 2

Vector ra has a magnitude of 49.0 m and points -20° from the positive x-axis, while vector rb has a magnitude of 75 m and points 45° from the positive x-axis. Analytical methods using trigonometric functions can determine their x- and y-components.

Step-by-step explanation:

The properties of vectors ra and rb in the context of the student's question include their magnitudes and the angles they make with the positive x-axis. For vector ra, which has a magnitude of 49.0 m and points 20° below the positive x-axis, we consider the angle to be -20° from the positive x-axis since it is below the axis. For vector rb, which has a magnitude of 75 m and points 45° above the positive x-axis, we use the angle as it is given, 45° from the positive x-axis. We use analytical methods to find the respective components along the x- and y-axes. To find the x- and y-components of each vector, you can use the following relationships: Ax = A cosθ and Ay = A sinθ, where A represents the magnitude of the vector and θ its direction angle with respect to the x-axis.