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    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.


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    (Original post by TSRforum)
    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.


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    not enough information
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    (Original post by TeeEm)
    not enough information
    Answers for a and b are:
    (5/2[sqrt(2)], 0, 5/2[sqrt(2)]) and 90 degrees


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    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.
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    (Original post by morgan8002)
    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 \alpha, \gamma angles made by \bold{r} with the x- and z-axes, and we have that the associated direction cosines satisfy \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1
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    (Original post by atsruser)
    There's definitely enough information - they have given values for the \alpha, \gamma angles made by \bold{r} with the x- and z-axes, and we have that the associated direction cosines satisfy \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1
    How would I use that equation for this question? I have never come across it.


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    This is an OCR question isn't it?


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    (Original post by TeeEm)
    not enough information
    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?
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    A hint for part b: try to draw the X y z axis, and have a line at pi/4 rad to X and y, what can said about its angle with z given this?


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    (Original post by TSRforum)
    How would I use that equation for this question? I have never come across it.


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    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 \pi/4 with the x-axis, then, when visualised as an arrow, its tip must lie on a right cone of angle \pi/4 along the x-axis. But it also makes an angle \pi/4 with the z-axis, so ...
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    (Original post by drandy76)
    A hint for part b: think about what it means to be perpendicular to I and k, and how would this relate to r and j?
    But it's not perpendicular to i and k.
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    (Original post by atsruser)
    But it's not perpendicular to i and k.
    Just re read it, edited now


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    I will only understand if some does a visual explanation or else it really does not make much sense to me.


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    (Original post by TSRforum)
    I will only understand if some does a visual explanation or else it really does not make much sense to me.


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    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.
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    "atsruser" gave a brilliant visual explanation. My own is according to "A slice of pi" hints.
    see attached file
    Attached Images
  1. File Type: pdf vectors.pdf (698.6 KB, 63 views)
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    (Original post by depymak)
    "atsruser" gave a brilliant visual explanation. My own is according to "A slice of pi" hints.
    see attached file
    Very nice solution
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    (Original post by depymak)
    "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.
 
 
 
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