In the adjacent figure, (i^cap, j^cap) represents a pair of unit vectors for the

Question

In the adjacent figure, (i^cap, j^cap) represents a pair of unit vectors for the ground-fixed coordinate system, while (b^cap_1, b^cap_2) represents a pair of unit vectors for a body-fixed coordinate system indicating the movement of a gauge that is attached to an airplane in motion. A third pair of unit vectors (c^cap_1, c^cap_2) is fixed with respect to the indicator needle within the gauge. Determine the coordinate transformation array between (b^cap_1, b^cap_2) and (c^cap_1, c^cap_2), and between (i^cap, j^cap) and (c^cap_1, c^cap_2).

Explanation / Answer

so from the figure we see

for transformation between (b1,b2) and (c1,c2)

-b1*sin(phi) + b2*cos(phi) = C1

-b1*cos(phi) - b2sin(phi) = c2

hence

[-sin(phi) cos(phi)][b1] = [c1]

[-cos(phi) -sin(phi)][b2] [c2]

so transformation matrix to go from (b1,b2) to (c1,c2) is

[-sin(phi) cos(phi)]

[-cos(phi) -sin(phi)]

for transformation form (i,j) to (c1,c2)

i*cos(theta+phi) + j*sin(theta+phi) = c1

-i*sin(theta+phi) + j*cos(t6heta+phi) = c2

so

[cos(theta+phi) sin(theta+phi)][i] = [c1]

[-sin(theta+phi) cos(theta+phi)][j] [c2]

hence transfromation matrix for transformation from (i,j) to (c1,c2) is

[cos(theta+phi) sin(theta+phi)]

[-sin(theta+phi) cos(theta+phi)]

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