Cross Product Geometry Problem

[Guest AdamFXB]

I've spent two hours trying to do this, but I cant figure a way out . We've just started the Gr. 12 Geometry and Discrete Mathematics and so far we've done dot product/cross product of vectors in 2,3 dimentions in cartesian and algebraic form

 

The sundial shown on the bottom works on the principle that the sun casts a shadow from a central projecting pointer onto the surface of a calibrated circle. The edge of the shadow is used to indicate the time of the day. A coordinate system can be placed on this object so that A is at the origin. A(0,0,0) and the coordinates of B and K are B(0,8,3) and C(-4,4,0).

 

Determine the following:

 

a) The magnitude of the vector projection of the edge of the pointer AB onto the calibration line AC

 

b) The size of Angle BAC

 

 

Any help would be greately appreciated

[TokenMonkey]

Have you been taught unit vectors? That's what you should use, as far as I can see.

[cronxeh]

You are given A(0,0,0) B(0,8,3) and C(-4,4,0).

 

1) AB dot AC

 

2) (AC)*(AB)*cos(BAC) = AB dot AC

 

AB = 8 j + 3 k

AC = -4 i + 4 j

AB = sqrt(8^2 + 3^2) = sqrt(64+9) = sqrt(73)

AC = sqrt((-4)^2 + 4^2) = sqrt(16 + 16) = sqrt(32)

 

AB dot AC = 32

or if you use unit vectors it should be.. 0.6621

 

1) 0.6621

2) angle BAC = cos^-1 ( 32/48.33) = 48.541