In the below vectors are represented in BOLD. Obviously you can’t do this with pen and paper so your vectors should be underlined; e.g. instead of not but .

For (a) (iii) we need to know the theorem

.

This reads, “vectors and are PERPENDICULAR if and only if the their dot product is zero.”

If and only means that the two statements are equivalent and the theorem actually says two things:

“IF the vectors are perpendicular, THEN their dot product is zero”, and

“IF the dot product of two vectors is zero, THEN they are perpendicular”.

To say the two statements — ‘the vectors are perpendicular’ and ‘the vectors have zero dot product’ — are equivalent means that whenever you hear one you can think of the other and vice versa.

So in our question here, if we are to suppose that ‘ is perpendicular to ‘ then by the above discussion this just means that

(*)

So to ask for the value of that makes the vectors perpendicular is the same as asking for the value of that makes (*) hold.

So we solve for :

Now calculating dot products is one of the easier things to do in this module:

I will let you finish this off.

Part (b) concerns WORK. We have that the work, , done by a force, , in moving on object through a displacement is given by

.

Here we are told that the displacement is and the force is .

Can you finish this off if I told you to watch the units?

Part (c) concerns the moment or TORQUE of a force. A force can have a turning effect and this turning effect is the moment or torque, . When the force, , acts at a displacement from the moment axis, , then the torque is given by

,

where ‘‘ is the VECTOR or CROSS PRODUCT.

At this point I can’t explain much without a picture and I would encourage you to look your notes.

Briefly the displacement is the vector that goes from the turning point to the point where the force acts and

.

The calculation of the cross product is also covered in your notes.

Regards,

J.P.

I am stuck on these vector questions that came up in the summer exam!

Question 2 (a) (iii), (b) and (c)

Thanks a million for your help.

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