Scalar product of vectors question

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October 12th 2009, 01:14 AM
Stuck Man

Scalar product of vectors question

I’ve studied vectors but I can’t do this question in a book.

PQRS is a rhombus of side 4 units. K, L, M and N are the mid-points of PQ, QR, RS and SP, respectively. SN® is a representative of vector u and SM® a representative of vector v. Show that SK®.SL® = 5u.v + 16.

The rhombus has P at the top left corner and S directly below. Q is slightly lower than P and R is directly below Q.
October 12th 2009, 10:18 AM
Opalg

Quote:

Originally Posted by Stuck Man

I’ve studied vectors but I can’t do this question in a book.

PQRS is a rhombus of side 4 units. K, L, M and N are the mid-points of PQ, QR, RS and SP, respectively. SN® is a representative of vector u and SM® a representative of vector v. Show that SK®.SL® = 5u.v + 16.

The rhombus has P at the top left corner and S directly below. Q is slightly lower than P and R is directly below Q.

If you draw a picture, you should be able to see that $\vec{SK} = 2\mathbf{u} + \mathbf{v}$ and $\vec{SL} = \mathbf{u} + 2\mathbf{v}$ . Also, $\mathbf{u.u} = \mathbf{v.v} = 2\times2 = 4$ . Now you just have to work out $(2\mathbf{u} + \mathbf{v})\mathbf{.}(\mathbf{u} + 2\mathbf{v})$ .
October 13th 2009, 03:06 AM
Stuck Man

I had got the first part. I didn't realise that the magnitude of u is 2 and the same for v. I suppose the answer can also be 36. Thanks.
October 13th 2009, 04:08 AM
Stuck Man

The scalar product of two vectors is the two magnitudes multiplied and multiplied by the cos of the angle (<=180). Why does the question not expect the result to be mutiplied by the cos of the unknown angle? So the answer should be 36 X cos of angle.
October 13th 2009, 04:18 AM
Opalg

Quote:

Originally Posted by Stuck Man

I had got the first part. I didn't realise that the magnitude of u is 2 and the same for v. I suppose the answer can also be 36. Thanks.

Quote:

Originally Posted by Stuck Man

The scalar product of two vectors is the two magnitudes multiplied and multiplied by the cos of the angle (<=180). Why does the question not expect the result to be mutiplied by the cos of the unknown angle? So the answer should be 36 X cos of angle.

I think you're confusing $\mathbf{u.u}$ with $\mathbf{u.v}$ . When you take the dot product of a vector with itself, the angle is 0, so you just get the square of the length of the vector. But when the vectors are different, as in $\mathbf{u.v}$ , that no longer applies. That is why the answer to the question leaves $\mathbf{u.v}$ as it is, without attempting to evaluate it. Unless you know the angle of the rhombus, you can't evaluate $\mathbf{u.v}$ .