Vector A has a magnitude of 20m and makes an angle of 30 degrees above the positive x axis. Vector B--> has a magnitude of 15m and is oriented 60 degree to the left of the y axis.

Question

Find the magnitude and direction of A--> - B-->. The magnitude and direction of 2A--> + b--> and the magnitude of -A--> + 3B-->

Answer 1

To find the magnitude and direction of A--> - B-->, we first need to find the components of both vectors A and B.

For vector A, the x-component can be found using cos(30°) = adjacent/hypotenuse = x/20m.
x = 20m * cos(30°) = 17.32m.

The y-component of vector A can be found using sin(30°) = opposite/hypotenuse = y/20m.
y = 20m * sin(30°) = 10m.

So, vector A--> can be represented as A--> = <17.32m, 10m>.

For vector B, we need to find the x and y-components given the orientation of 60 degrees to the left of the y-axis.

The angle between the y-axis and the vector B is 90° - 60° = 30°.
The x-component of vector B is then 15m * cos(30°) = 15m * 0.866 = 12.99m.

The y-component of vector B is -15m * sin(30°) = -15m * 0.5 = -7.5m (negative because it is oriented downward from the y-axis).

So, vector B--> can be represented as B--> = <12.99m, -7.5m>.

To find the magnitude and direction of A--> - B-->, we subtract the corresponding components:

A--> - B--> = <17.32m - 12.99m, 10m - (-7.5m)> = <4.33m, 17.5m>.

The magnitude of A--> - B--> is √(4.33^2 + 17.5^2) = 18.07m.

To find the direction, we use tan^-1(17.5m/4.33m) = 76.33° above the positive x-axis.

Therefore, the magnitude of A--> - B--> is 18.07m and the direction is 76.33° above the positive x-axis.

To find the magnitude and direction of 2A--> + B-->, we multiply vector A--> by 2 and then add vector B-->:

2A--> + B--> = 2<17.32m, 10m> + <12.99m, -7.5m> = <34.64m, 20m> + <12.99m, -7.5m> = <47.63m, 12.5m>.

The magnitude of 2A--> + B--> is √(47.63^2 + 12.5^2) = 49.27m.

To find the direction, we use tan^-1(12.5m/47.63m) = 15.1° above the negative x-axis.

Therefore, the magnitude of 2A--> + B--> is 49.27m and the direction is 15.1° above the negative x-axis.

To find the magnitude of -A--> + 3B-->, we multiply vector A--> by -1 and vector B--> by 3 and then subtract:

-A--> + 3B--> = -1<17.32m, 10m> + 3<12.99m, -7.5m> = <-17.32m, -10m> + <38.97m, -22.5m> = <21.65m, -32.5m>.

The magnitude of -A--> + 3B--> is √(21.65^2 + (-32.5)^2) = 39.16m.

Therefore, the magnitude of -A--> + 3B--> is 39.16m.