Finding it a little troublesome doing these three questions, I'm sure the first one's relatively straightforward but I seem to be having a bit of trouble . Can anyone give me a couple hints? Cheers.
1) Simplify a.ba×b
2) Show that the foot of the perpendicular from the point p to the line r=a+λu has position vector:
a+∣u∣2u.(p−a)u
3) A high energy particle beam is fired at a sphere of matter. The beam is initiated from point a in a direction d at a sphere, centred on the origin, with radius p. What must the radius of the sphere be in terms of a and d in order for there to be a collision with the energy beam?
Looking at the first one, I'm not sure what they're trying to achieve. If we go to the definitions of the dot and cross products this reduces to (tanθ)n.
But the question arises as to how to specify n and tanθ and the simplest method as far as I can see is with the original formula.
2) Show that the foot...
Hint: What do you know about perpendicular vectors?
2) Show that the foot of the perpendicular from the point p to the line r=a+λu has position vector:
a+∣u∣2u.(p−a)u
1. Sketch the line r=a+λu, showing vectors a,u
2. Draw a position vector p somewhere in the plane.
3. Draw the perpendicular vector s, say, from p to the line. Call the point where they meet F. Note that the position vector of F is rF=a+λpu for some scalar λp which depends on p.
Your job is to find an expression for λp in terms of quantities you already know.
4. Let the vector along the line from a to F be t. Write down an equation relating s,a,p,t. (Hint: draw a suitable triangle, one of whose sides is t).
5. Note that we can write t=λpu
6. Write down in vector notation the condition that s is perpendicular to the line.
Using the result of question 2, what's the position vector for the foot of the perpendicular from the origin to the line?
Then just find its length. This will be the minimum radius to intersect the trajectory. I.e. the path will be a tangent to the sphere, the perpendicular being a radius.
Using the result of question 2, what's the position vector for the foot of the perpendicular from the origin to the line?
Then just find its length. This will be the minimum radius to intersect the trajectory. I.e. the path will be a tangent to the sphere, the perpendicular being a radius.
Sorry I'm just really baffled by this question, it seems so straightforward but I can't put my mind to it
Am I right in saying the position vector is a−∣d∣2a.dd?