Degree level vector help required.

[Brains_Tom]

Hi folks first post, i am a 20yr old student from north-west england, hope you guys can help me and i can return the favour some time.

1) How can the vector product be used to show two vectors are orthogonal to each other?
a=2i-5j+3k
b=-i+j+k

2)How can the vector product be used to show an expression for the area of a triangle using the below co-ordinates?
(2,0,1) (3,1,7) (-2,5,5)

3)Using two lines...
L1: r=p+λn L2: s=q+µm
where...
p=8i+9j+3k
n=i+j
q=-2i-4j+k
m=i+2j

...and a suitable minimum distance formula, find the shortest distance between the two lines, providing a straight line indicates this distance, find the position vectors of these two connecting points, one on each line.

Many thanks.

 

[AlphaNumeric]

1) $$|u \times v| = |u||v|\sin \theta$$

2) Use http://mathworld.wolfram.com/SASTheorem.html

3) I forget how to do this one, it's been a long time.

 

[Brains_Tom]

I know what the vector product is mate i just don't know how to answer the question, as for 2, can you clarify, how do i put this into three dimensions for an actual result?

 

[AlphaNumeric]

Well if 2 vectors are othogonal, you'll get $$\sin \theta=1$$. Then you just show that $$|u \times v| = |u||v|$$, which just involves you sitting there and working it out manually.

For the second question, you're given the position of the triangles vertices, so you can derive vector expressions for the sides. Then, you can see from the link I posted that the area of a triangle is the half the product of 2 sides times sin of the angle between them. Sound familiar, its
$$\frac{1}{2}|u||v|\sin \theta$$ which is otherwise written $$\frac{1}{2}|u \times v|$$. Hence, work out the vector expressions for two of the triangles sides and then halve the mod of their cross product for the area.

For the third one you're looking for a plane which one line lies in and which is normal from the second one.

 

[Brains_Tom]

1) But i get mod(a) x mod(b) = sqrt(114) and mod(axb) = sqrt(38) ?

 

[Physics Monkey]

If you look closely you'll see that they aren't actually orthogonal to each other. Try calculating the dot product for example. If that -1 in front of the i on b was a 1, then the vectors would be orthogonal.

 

[Brains_Tom]

SORRY - EDIT

1) How can the vector product be used to show two vectors are orthogonal to each other?
a=2i-5j+3k
b=-i+j+k

2)How can the vector product be used to show an expression for the area of a triangle using the below co-ordinates?
(2,0,1) (3,1,7) (-2,5,5)

3)Using two lines...
L1: r=p+λn L2: s=q+µm
where...
p=8i+9j+3k
n=i+j
q=-2i-4j+k
m=i+2j

...and a suitable minimum distance formula, find the shortest distance between the two lines, providing a straight line indicates this distance, find the position vectors of these two connecting points, one on each line.

Many thanks.

1) Use the vector product to find two unit vectors that are orthogonal to both 'a' and 'b'? a=2i-5j+3k b=-i+j+k

2)How can the vector product be used to show an expression for the area of a triangle using the below co-ordinates?
(2,0,1) (3,1,7) (-2,5,5)
Find the area by evaluating this vector product expression.

Well annoying, sorry.

 

[Physics Monkey]

Ok, the question makes a good deal more sense now. The basic fact to use is that the vector product produces a vector which is orthogonal to both the inputs.

 

[Brains_Tom]

Ok, the question makes a good deal more sense now. The basic fact to use is that the vector product produces a vector which is orthogonal to both the inputs.

But what do i use for the angle, do i presume sin(theta) is just one, then just multiply the magnitudes, surely that would just give a numeric answer, when i am after a vector, i do understand that my two answers will be the same vector just opposite sign though.