i, j and k are unit vectors in the x-, y- and z-directions respectively. The three-dimensional vector r has magnitude 5, and makes angles of 1/4(pi) radians with each of i and k.
(A) Write r as a column vector, leaving any square roots in your answer
(B) State the angle between r and the unit vector j.
Don't really know how to use info given in question to answer parts a and b.
i, j and k are unit vectors in the x-, y- and z-directions respectively. The three-dimensional vector r has magnitude 5, and makes angles of 1/4(pi) radians with each of i and k.
(A) Write r as a column vector, leaving any square roots in your answer
(B) State the angle between r and the unit vector j.
Don't really know how to use info given in question to answer parts a and b.
Think about it geometrically. I disagree witb TeeEm; r is uniquely defined. Consider k and i and where r would be in relation to those.
There's definitely enough information - they have given values for the α,γ angles made by r with the x- and z-axes, and we have that the associated direction cosines satisfy cos2α+cos2β+cos2γ=1
There's definitely enough information - they have given values for the α,γ angles made by r with the x- and z-axes, and we have that the associated direction cosines satisfy cos2α+cos2β+cos2γ=1
How would I use that equation for this question? I have never come across it.
If you let r = ai + bj + ck, then using the cosine of angle formula in terms of dot products etc gives a and c, and then we can work out b using the given magnitude of the vector, or am I missing something?
If you haven't come across it, then don't worry. You can solve the problem via geometric intuition: if the vector r makes an angle π/4 with the x-axis, then, when visualised as an arrow, its tip must lie on a right cone of angle π/4 along the x-axis. But it also makes an angle π/4 with the z-axis, so ...
Did you understand my description of the two cones that the tip of the vector must lie on? If not, try drawing out the xyz-axes and start to mark possible positions for the tip of the vector, given the angles that it must make with the axes. You should be able to see that there is only one possible location that makes the correct angle with both axes.
"atsruser" gave a brilliant visual explanation. My own is according to "A slice of pi" hints. see attached file
You've used the direction cosine approach that I suggested upthread (though you seem to have derived it from scratch) but note that the question says "State the angle .." - that's usually a big hint that extensive calculation is not expected, so I guess that they're looking for some quick and dirty justification at most.