Express the vector A = 3i + 2j + k in terms of the triad i j k\', where the x\'

Question

Express the vector A = 3i + 2j + k in terms of the triad i j k', where the x' y' -axes are rotated 45 degree around the z-axis, with the z- and z'-axes coinciding, as shown in Figure 1.8.1. Refferring to the figure, we have for the coefficients of transformation i.i = cos 45 degree and so on; hence, i.i' = 1/squareroot 2 j.i' = 1/squareroot 2 k.i' = 0 These give A_x' = 3/squareroot 2 + 2/squareroot 2 = 5/squareroot 2 Ay' = -3/squareroot 2 + 2/squareroot 2 = -1/squareroot 2 A_z' = 1 so that, in the primed system, the vector A is given by A = 5/squareroot 2 I' - 1/squareroot 2 j' + k'

Explanation / Answer

i.ii=i.ii cos 45=1X1 X 1/sqrt2

hence in a dot product like this we should multiply magnitude of one vecor X magnitude of another vectorX angle between the vectors.

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