Some vectors and buried treasure.

[niyati]

I have two questions:

1. Vector A has x and y components of -8.70 cm and 15.0 cm, respectively; vector B has x and y components of 13.2 cm and -6.60 cm, respectively. If A - B + 3C = 0, what are the components of C?

To start out this problem, I calculated A - B :

[-8.70,15.0] - [13.2,-6.60] = [-21.9,21.6]

and then replaced it in the entire equation in terms of i and j:

(-21.9i + 21.6j) + 3([Cx]i + [Cy]j) = 0

The follow-through:

3([Cx]i + [Cy]j) = 0 - (-21.9i + 21.6j)

3([Cx]i + [Cy]j) = (0i + 0j) - (-21.9i + 21.6j)

3([Cx]i + [Cy]j) = (21.9i - 21.6j)

Now, I heard that you could not divide a vector quantity by a scalar quantity, but would it be all right to multiply the other side by one-third and then match up the quantities?

([Cx]i + [Cy]j) = (1/3)(21.9i - 21.6j)

Cx = 7.3 cm
Cy = -7.2 cm

2. Instruction for finding a buried treasure include the following: Go 75.0 paces at 240 degrees, turn to 135 degrees and walk 125 paces, then travel 100 paces at 160 degrees. The angle are measure counterclockwise from an axis point to the east, the + x direction. Determine the resultant displacement from the starting point.

...I am very confused. I suppose counter-clockwise from the east means something like a coordinate grid/polar coordinate grid. But, (and this might seem like a weird conceptualization) with each stopping point, before each turn and after the pacing, is it like a new coordinate grid? Like, for example, after I go for 75 paces at a 240 degree angle, do I then draw another imaginary coordinate grid and go for 135 degrees? Or, do I add 240 to 135 (which is 375, or 15 degrees) and go 15 degrees? It's kind of a stupid question but, well, I think once I get that straight, I'm good to go.

 

[Doc Al]

#1

Now, I heard that you could not divide a vector quantity by a scalar quantity, but would it be all right to multiply the other side by one-third and then match up the quantities?

You may certainly divide a vector by a scalar. (Multiplying by 1/3 is the same as dividing by 3.)

#2

But, (and this might seem like a weird conceptualization) with each stopping point, before each turn and after the pacing, is it like a new coordinate grid? Like, for example, after I go for 75 paces at a 240 degree angle, do I then draw another imaginary coordinate grid and go for 135 degrees?

Yes. Giving the angle is equivalent to giving the direction in terms of north/south/east/west.

Or, do I add 240 to 135 (which is 375, or 15 degrees) and go 15 degrees?

No. When you are walking at 240 degrees that's one direction with respect to the x-axis; when you are walking at 135 degrees that's another direction. Treat each segment independently and then add their components to find the resultant.

 

[niyati]

For #1, that's exactly what I thought. But I remember some rule against dividing and vectors. Might it have been a vector by a vector?

And for #2, thank you!

 

[Doc Al]

But I remember some rule against dividing and vectors. Might it have been a vector by a vector?

Yep. That's the one you can't do.