1. ## Vector help

Okay, so I've just started Engineering Mathematics and funnily enough, we get to start off with vectors, something I've never used before. To make matters worse, my teacher mumbles into the whiteboard so I can neither hear what he says nor understand the jargon he writes.

We also have weekly assignments and I'm looking at this week's with a blank face. I managed to decipher vector products without too much trouble, but the rest is leaving me confused.

Hope you guys can help.

Okay, the first question looks like algebra, which means it probably isn't:
1) If a = 6j + 2k, b = 4i - 5j and c = -i + j +2k
find 2a - 4b +c

The second question talks about finding the right direction to move in.
2) A man wished to row across a lake to a point directly south of him. If the lake has a current flowing west at 3km/hr and he can row at 7km/hr, find the direction in which he must row.

The third question talks about vector components.
3) Fine the (vector) component of 3i - j +5k in the direction of -i + 4j - 2k

The fourth question was about vector products, which I'm pretty sure I have a good grasp on. Still, an elaboration on how to work them out would be nice, just so I can be sure I've done it right.
4) Find the vector product:
(2i - j) x (6i + 3j + 2k)

I've got:
let a = 2i = j, b = 6i + 3j + 2k

a * b =
| i j k |
| 2 1 0|
|6 3 2|

=
i| 1 0 | - j| 2 0 | + k| 2 1 |
.| 3 2 | .. | 6 2 | .... | 6 3 | (ignore the dots)

=
i( (1)(2) - (0)(3) ) - j( (2)(2) - (0)(6) ) + k( (2)(3) - (1)(6) )

= 2i - 4j

----

Any help would be great, thanks.

2. Originally Posted by fantanoice
...
Okay, the first question looks like algebra, which means it probably isn't:
1) If a = 6j + 2k, b = 4i - 5j and c = -i + j +2k
find [I]2a - 4b +c

...
The components (i,j,k) of a vector in 3D-space correspond to the three axes of a coordinate system.

If a = 6j + 2k = 0i + 6j + 2k then 2a = 12j + 4k
If b = 4i - 5j = 4i - 5j + 0k then -4b = -16i + 20j
If c = -i + j +2k then c = -i + j +2k

Now add columnwise and collect like terms:

2a - 4b +c = -17i + 33j + 6k

3. Originally Posted by earboth
The components (i,j,k) of a vector in 3D-space correspond to the three axes of a coordinate system.

If a = 6j + 2k = 0i + 6j + 2k then 2a = 12j + 4k
If b = 4i - 5j = 4i - 5j + 0k then -4b = -16i + 20j
If c = -i + j +2k then c = -i + j +2k

Now add columnwise and collect like terms:

2a - 4b +c = -17i + 33j + 6k
Wow, that's the answer I got simply by substituting a, b and c into the equation.

I assumed it would be more complex than that. Thanks.

Anybody know how to solve 2 and 3?

4. Originally Posted by fantanoice
...
[/I]The second question talks about finding the right direction to move in.
2) A man wished to row across a lake to a point directly south of him. If the lake has a current flowing west at 3km/hr and he can row at 7km/hr, find the direction in which he must row.

Draw a rough sketch: You'll get a right triangle with one leg pointing South and one leg pointing $East$ because the man has to componsate the current running West. The course is described by the hypotenuse. Use the Sine function to calculate the angle of deviation from South.
...

The fourth question was about vector products, which I'm pretty sure I have a good grasp on. Still, an elaboration on how to work them out would be nice, just so I can be sure I've done it right.
4) Find the vector product:
(2i - j) x (6i + 3j + 2k)

I've got:
[I]let a = 2i = j, b = 6i + 3j + 2k

a * b =
| i j k |
| 2 -1 0| <<<<<< there is a negative sign missing
|6 3 2|

...
=
i( (-1)(2) - (0)(3) ) - j( (2)(2) - (0)(6) ) + k( (2)(3) - (-1)(6) )

= -2i - 4j + 12k

...
...

5. Okay, so I was using the right function for 4, just messed up a sign. Okay.

Thanks for 2, I'll give it a go.