A cube is 3 cm on a side, with one corner at the origin. What is the unit vector pointing from the origin to the diagonally opposite corner at location<3,3,3>cm? What is the angle from this diagonal to one of the adjacent edges of the cube?

Short Answer

Expert verified

The unit vector is:r^=0577,0.577,0577

The angle from the diagonal to the adjacent edges of the cube:

θx=θy=θz=θ=54.76°

Step by step solution

01

Identification of given data

Side of cube = 3cm

Diagonally opposite corner at 3,3,3

02

Calculating unit vector

To find the vector that points from the origin to the opposite corner of the cubesubtract the initial location from the final location

r=3,3,3-0,0,0=3,3,3cm

The magnitude of this vector is

r=32+32+32=5.196cm

The unit vector for that vector is

r^=rr=3,3,35.196=0577,0.577,0577

03

Calculating angles from the diagonal vector

Recall that the unit vector could be given in terms of the direction cosines

r^=cosθx,cosθy,cosθz

which means:

cosθx,cosθy,cosθz=0577,0.577,0577

therefore

cosθx,cosθy,cosθz=0.577

So, the angle from the diagonal vector to one of the adjacent edges is ( n.b.the three adjacent edges are on thex,y,and zaxis since the side that the origin

θ=cos-10577θ=54.76°

Where,θx=θy=θz=θ

Therefore, the required vector isr^=0577,0.577,0577and the angle from the diagonal to the adjacent edges of the cube isθx=θy=θz=θ=54.76°.

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