# Two Vector Questions

• May 25th 2008, 06:47 AM
Jeavus
Two Vector Questions
1. A pilot wishes to fly from Toronto to Montreal, a distance of 508km on a bearing of 075°. The cruising speed of the plane is 550km/h. An 80km/h wind is blowing on a bearing of 125°.
a) What heading should the pilot take to reach his destination?
b) What will be the speed of the plane relative to the ground?
c) How long will the trip take?

2. The dot product has a geometric interpretation as an area. Let vector a = vector OA and vector b = vector OB be any two vectors forming an angle theta. Then vector a (dot product) vector b can be expressed as |a||b|cos(theta).
a) Assume that 0° < theta < 90°. On the diagram below, point C was constructed so that angle OCB = 90°. Then point D was constructed so that OD = OC. Segments OA and OD form adjacent sides of the rectangle OAED. Show that vector a (dot product) vector b represents the area of this rectangle.
b) Describe what happens to the rectangle when theta = 90° and when theta = 0°.
c) What special case occurs when theta = 45°? Explain.

http://img389.imageshack.us/img389/1557/diagram3tz4.png
I've looked at both of these questions and I cannot seem to get the correct answers that are being given. If anyone could provide any insight I'd be thankful. :)
• May 25th 2008, 07:06 AM
Moo
Hello,

Quote:

Originally Posted by Jeavus
2. The dot product has a geometric interpretation as an area. Let vector a = vector OA and vector b = vector OB be any two vectors forming an angle theta. Then vector a (dot product) vector b can be expressed as |a||b|cos(theta).
a) Assume that 0° < theta < 90°. On the diagram below, point C was constructed so that angle OCB = 90°. Then point D was constructed so that OD = OC. Segments OA and OD form adjacent sides of the rectangle OAED. Show that vector a (dot product) vector b represents the area of this rectangle.
b) Describe what happens to the rectangle when theta = 90° and when theta = 0°.
c) What special case occurs when theta = 45°? Explain.

http://img389.imageshack.us/img389/1557/diagram3tz4.png
I've looked at both of these questions and I cannot seem to get the correct answers that are being given. If anyone could provide any insight I'd be thankful. :)
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Let's study this problem.

a) Area of the rectangle : $\displaystyle A$

$\displaystyle A=OD \times OA=OC \times |\vec{a}|$, because OC=OD and because OA is the length of vector a.

Calculating OC :
Consider the right angle triangle OBC.

Is there a formula that makes a relation between $\displaystyle \theta, \ OC, \text{ and } OB=|\vec{b}|$ ? :)

b) If theta=90°...
But I think there is an information missing about how C is constructed... C has to be on OA.
If so, then C will be O.
Therefore, OC=0.

Thus OD=0.

Try to work it out :)
• May 25th 2008, 07:12 AM
Jeavus
Indeed, cos(theta) relates theta, OC, and OB.

How does this prove that the dot product of vector a and vector b represents the area though?
• May 25th 2008, 07:14 AM
Moo
Quote:

Originally Posted by Jeavus
Indeed, cos(theta) relates theta, OC, and OB.

How does this prove that the dot product of vector a and vector b represents the area though?

Well,

$\displaystyle A=OC \times |\vec{a}|$

$\displaystyle \cos \theta=\frac{OC}{OB} \implies OC=OB \cdot \cos \theta=|\vec{b}| \cdot \cos \theta$

$\displaystyle \implies A=|\vec{b}| \cdot \cos \theta \times |\vec{a}|$

But we know that $\displaystyle \vec{a}.\vec{b}=|\vec{a}|\cdot |\vec{b}| \cdot \cos \theta$

What do you conclude ? :p
• May 25th 2008, 07:17 AM
Jeavus
Oh, haha. That was rather simple and I didn't even see it.

Thanks a lot!
• May 25th 2008, 08:34 AM
Jeavus
Anyone have any idea for #1?
• May 25th 2008, 06:02 PM
Jeavus
I've solve for the heading in 1a, but I need help with the rest.

Can anyone help? :)