Find Vector Magnitude 27: Algebraic Vectors

[masterofthewave124]

if i had to find a vector of magnitude 27 units which is parallel to 3i+ 4j, what do i have to do first? would i express my answer in ordered pair notation?

 

[Hootenanny]

if i had to find a vector of magnitude 27 units which is parallel to 3i+ 4j, what do i have to do first? would i express my answer in ordered pair notation?

I would express your answer in terms of unit vectors. Ordered pair notation is uaually used to define co-ordinates. Remember parallel vectors must be multipuls of each other.

~H

 

[masterofthewave124]

would this be a sufficient answer:

let u be the vector.

the magnitude of 3i + 4j is 5

27/5 = 5.4

since parallel vectors are multiples of each other, we can just multiply (3i + 4j) by the scalar 5.4

so u = 5.4(3i + 4j)
= 16.2i + 21.6j

this is what i came up with before but it seems messy (decimals) and and kind of an inefficient solution

 

[arildno]

Seems good to me, but why not write your answer as:
$$u=\frac{27}{5}(3i+4j)$$

Note that $\frac{1}{5}(3i+4j)$ is a UNIT vector.

 

[masterofthewave124]

looks much better. thanks!

 

[Hootenanny]

Note that $\frac{1}{5}(3i+4j)$ is a UNIT vector.

Who said it wasn't a unit vector?

 

[arildno]

Who said it wasn't a unit vector?

Noone. I just wanted to emphasize that, so that OP could see how a unit vector naturally would occur in his expression.

 

[Hootenanny]

Noone. I just wanted to emphasize that, so that OP could see how a unit vector naturally would occur in his expression.

Ahh right, I thought I'd made a mistake somewhere .

 

[arildno]

No, but I wanted OP to see why your suggestion

"I would express your answer in terms of unit vectors."

is, indeed, the most natural one.

 

[Hootenanny]

No, but I wanted OP to see why your suggestion

is, indeed, the most natural one.

Good point, I should have emphisised (but I didn't get the chance )

 

[masterofthewave124]

guys, i have some more questions if you don't mind.

state whether the following expressions are vectors, scalars or meaningless:
a) (a+b) • (a+c)
b) (a • a)b
c) (a • b)• c(b × c)

note that all the letters are vectors (so picture arrows on top of them)

mostly i need to clairify some things. first, what's the difference between a scalar and something meaningless? also for b) and c), what does something like (a • a)b compute to. does the b multiply, or dot, or cross or what?

 

[arildno]

Something "meaningless" is an indicated operation between quantities that the operation in question cannot, by definition, be used upon.

 

[masterofthewave124]

oh ok, so since the dot and cross products are only applicable to vectors, if a scalar was involved in one of them, the result would be meaningless?

to restate my other question, what does the b part of (a • a)b compute to. does the b multiply, or dot, or cross or what?

 

[arildno]

Remember that you have 3 basic product operations with vectors:
1. Dot product: Takes two vectors and produces a scalar.
2. Cross product: Takes two vectors and produces a vector.
3. Scalar multiplication: Takes a scalar and a vector, produces a vector that is parallell to the original vector.

 

[masterofthewave124]

thanks for the summary but the way (a • a)b is written, and considering b is a vector and not a scalar, what is the operation here?

 

[arildno]

What does a parenthesis usually signify for the order of operations performed?

 

[masterofthewave124]

so i think your referring to doing the inside of the brackets first; a dot product operation yielding a scalar. so then does it leave a scalar multiplying a vector?

 

[arildno]

What do YOU think?

 

[masterofthewave124]

lol that's what i think, i do have some doubt however because as you look at c) (a • b) • c(b × c), the second part of the question (c(b × c)) is somewhat confusing. you basically get a vector multiplying a vector without dot or cross product operation (the only two ways you can muliply vectors I'm aware of). although this does not change the outcome of the question since the first part of the question is a scalar and yielding the result meaningless.

 

[arildno]

Correct!
Either way you are looking upon the last one, either as:
$$((a\cdot{b})\cdot{c})(b\times{c})$$
or as:
$$(a\cdot{b})\cdot({c}(b\times{c}))$$
the expressions are meaningless.

 

[teken894]

thanks for the summary but the way (a • a)b is written, and considering b is a vector and not a scalar, what is the operation here?

(a • a) would result in a scalar value

hence that scalar result would then have to be multiplied with the vector b (scalar multiplication)

The final answer is a vector

 

[masterofthewave124]

another question...

A parallelogram PQRS has vertices P(1, 3, 1), Q(4, 5, −2), S(−2, 3, −5). Determine
a) the coordinates of R
b) the perimeter of the parallelogram
c) the area of the parallelogram

if it was in 2-d if would be so much better but this is so hard to visualize!

for a) i know i have to come up with an equation like PQ = RS, but then what do i do from there. do i add P and Q (R and S), subtract or what?

 

[Hootenanny]

You must first find the vector equation of the line PQ.

~H

 

[arunbg]

You need to specify that the vertices are taken in order as well, otherwise there are three possible values for the coordinates of R.

You can also approach the problem non vectorially .
What is the property of the point of intersection of diagonals of a parallelogram ?
For the area bit, I'd say vectors are cool .

 

[masterofthewave124]

Hootenanny: we haven't learned vector equations...i think were supposed use vector operations (adding and subtracting) to solve for R.

arunbg: what order are you talking about?

 

[arildno]

Note that if S-P=R-Q, then you have R-S=Q-P.
See how you may translate this idea into something useful.

 

[Hootenanny]

Another little hint, think of how you can get to R using the vectors you have been given (this involves adding and subtracting as you say)

 

[arunbg]

arunbg: what order are you talking about?

The order that I am talking of is the order in which you take the vertices.
Suppose you are given three non collinear points on a piece of paper and you are asked to find out the fourth point such that the figure formed by joining the points is a parallelogram . If the order in which you join the points (say P to Q, Q to R ,R to S and S to Q) is not specified, then there are 3C1 = 3 possible locations for R. Do this on a paper and you can see why .In this question you have assumed PQRS to be the order.

In my earlier post, I was referring to the property that diagionals of a parallelogram bisect each other or the midpoints of the two diagonals coincide . Remember section formula ?


Or of course, for vector method, what is the pallelogram law of vector addition ?

 

[masterofthewave124]

ok its been a while but I'm only coming back to this question now.

taking the vertices in order, i found the coordinates of R to be (1,5,-8)

for b), would the perimeter be equal to 2(|PQ| + |QR|)?

for c), how do i find the height? would it be |RS x RQ|?

 

[arunbg]

b), would the perimeter be equal to 2(|PQ| + |QR|)?

Yes that wouls be correct .

c), how do i find the height? would it be |RS x RQ|?

No, the height would be
$$\vert\vec{RS}\times\widehat{RQ}\vert$$

where $\widehat{RQ}$ is a unit vector along RQ .
Do you see why ?
As a side note, the area of the parallelogram can be directly found using
|RS x RQ| .