Displacement rightarrow d_1 is in the yz plane 55.2 degree from the positive dir

Question

Displacement rightarrow d_1 is in the yz plane 55.2 degree from the positive direction of the y axis, has a positive z component, and has a magnitude of 5.19 m. Displacement rightarrow d_2 is in the xz plane 24.6 degree from the positive direction of the x axis, has a positive z component, and has magnitude 1.86 m. What are (a) rightarrow d_1 middot rightarrow d_2, (b) the x component of rightarrow d_1 times rightarrow d_2, (c) the y component of rightarrow d_1 times rightarrow d_2, (d) the z component of rightarrow d_1 times rightarrow d_2 and (e) the angle between rightarrow d_1 and rightarrow d_2? Number Units Number Units Number Units Number Units Number Units

Explanation / Answer

First you need to calculate the x, y and z components for each vector using trigonometry.
For instance, d1 projects on to the:
y-axis at 5.19 * cos(55.2) = 2.96.
x-axis at 0 (the d1 vector is in the yz plane, so no x component)
z-axis at 5.19 x sin(55.2) = 4.26.
The d1 vector can now be written as d1 = [0 4.26 2.96] i.e [x y z].
Calculating the d2 vector x,y,z components gives d2 = [1.69 0 0.77]

Now, d1.d2 is the scalar product. The answer is just a number, not a vector.
d1.d2 = (0 x 1.69) + (4.26 x 0) + (2.96 x 0.77) = 2.28 m (Answer).

d1xd2 is a little trickier to show. Write a matrix with d1 on the top row, and d2 on the bottom row, with the columns [x y z]
| x___    y __    z |
| 0    4.26    2.96 |
| 1.69    0    0.77 |

Now, cover the x column and calculate x = (4.26x0.77) - (2.96x0) = 3.28 m.
Now, cover the y column and calculate y = - ( (0x0.77) - (2.96x1.69)) = 5.00 m (Notice the minus symbol)
Now, cover the z column and calculate z = (0x0) - (4.26x1.69) = -7.2 m.
(Look up 'Determinant')

So, d1 x d2 = [3.28 5.00 -7.2] where each number represents the projection of the vector on to the x, y and z axes respectively.

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