# Vector

#### Indranil

If the vector A = 2iˆ+4jˆ and B = 5iˆ+pjˆ are parallel to each other, the magnitude of B is____________
how to solve it?

#### HallsofIvy

In order to be parallel to A, B must be a multiple of A. That is 5i+ pj= c(2i+ 4j)= 2ci+ 4cj. 5i= 2ci so what is c? Then what is p? So what is the length of 5i+ pj?

#### Indranil

In order to be parallel to A, B must be a multiple of A. That is 5i+ pj= c(2i+ 4j)= 2ci+ 4cj. 5i= 2ci so what is c? Then what is p? So what is the length of 5i+ pj?
C= 5/2, P = 10, the length is 5i + 10j
still, I don't understand the concept 'In order to be parallel to A, B must be a multiple of A' Could you make this part a little bit easier with easy example?

#### benit13

C= 5/2, P = 10, the length is 5i + 10j
still, I don't understand the concept 'In order to be parallel to A, B must be a multiple of A' Could you make this part a little bit easier with easy example?
In vector mathematics, vectors have a magnitude and a direction. In general, we have

$$\displaystyle v = a \hat{r}$$

where $$\displaystyle a$$ is a scalar number and $$\displaystyle \hat{r}$$ is a vector that describes the direction. The convention is to use a direction vector with length 1 (which is what the little hat means above the r), but this need not always be the case.

Let's look at some examples.

If you take a piece of paper and plot the following vectors (start at the origin):

4i
-4i
4j
-4j

You'll see that they point in different directions, but they have the same length. In these cases, $$\displaystyle a$$=4 but the direction vectors change (i, -i, j, -j).

If you take a new piece of paper and plot the following vectors:

2i+j
4i+2j
6i+3j
8i+4j
10i + 5j

You'll see that they all point in the same direction (parallel), but they are successively longer and longer vectors.

For these vectors, $$\displaystyle a$$ is equal to 1, 2, 3, 4 or 5 and 2i+j is describing the direction. So we can write:

$$\displaystyle v = a (2i +j)$$

If we choose to use the convention, we describe the vector instead using

$$\displaystyle v = a \left(\frac{2}{\sqrt{5}}i + \frac{1}{\sqrt{5}}j\right)$$

but I wouldn't worry about that for now.

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#### HallsofIvy

C= 5/2, P = 10, the length is 5i + 10j
still, I don't understand the concept 'In order to be parallel to A, B must be a multiple of A' Could you make this part a little bit easier with easy example?
It looks like you don't know what a vector is! 5i+ 10j is a vector, not the "length of a vector". The length of a vector is a number. The length of the vector 5i+ 10j is $\sqrt{5^2+ 10^2}= \sqrt{25+ 100}= \sqrt{125}= \sqrt{(25)(5)}= 5\sqrt{5}$.

#### Woody

Hands On Approach

Some people find it difficult to link the pure abstraction of the mathematical formula
with the actual (physical) meaning of vectors.

In my opinion there is nothing like getting out the old pencil and ruler
and drawing the vectors out on graph paper.

It might seem anachronistic, in these days of computer graphics packages, of Excel and MatLab, etc...

But I believe that the physical act of creating lines on the paper
that demonstrate the geometric issues that the maths is mapping
can help to develop a "feel" for the way vectors work.

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