Question

I always took components of vectors to be vectors. But now my book says "component of vectors are not vectors. They are scalars."

My understanding:
They have both direction and magnitude and on vector addition they give the original vector and yet my book says they are scalars.

Can anyone explain how?

Answer 1

they are originally vectors.
a scalar addition can't give a vector.
so book may be wrong @ this point.

Answer 2

The text book which says so is "University Physics with Modern physics, 12th edition" by HUGH D. YOUNG
ROGER A. FREEDMAN

One thing which gives me a hint that they might be scalars is that components are essentially projections! which might give a conclusion that they are scalars! But then i am not sure.

Answer 3

ya I agree that they are projections.
but they are basically vectors.
& I also have that book.
yet there are some exceptional cases such as components of vectors,
they are basically vectors which gives out a vector but can be also termed as scalar becoz they are only projections on x-axis & does not have any matter about real world.
we use them as they are the solutions of many questions in physics.
but we can't give them a title of their real existence as a vector exists.
so as a summary, they are vectors.
becoz the majority of all physics book says they are vectors.

Answer 4

@maheshmeghwal9 so, your conclusion is that they are vectors because majority of physics textbook says they are vectors..

Answer 5

ya absolutely.

Answer 6

by the way on which page no. this quotation is written in HD Young?

Answer 7

There was a time when every single human on earth said earth is flat.

Answer 8

Page-36

Answer 9

k! wait for 2 min.

Answer 10

So this is what u are saying about?

Answer 11

?

Answer 12

yep. The part in the caution.

Answer 13

k! think on that:)
If we walk through the book according to that caution.k!
then it is that thing which I was saying earlier that they are only projections. so termed as scalar
they does not have right to matter with real world as a vector has that right.
but how can u calculate a vector with 2 scalars?
can u tell me?
It is impossible.
so u must take the summary as they are "scalar vectors" which r unreal but play an important role in calculating vectors.

Answer 14

sorry but I can say only this much:)

Answer 15

If u have doubt in my reply then ask me:)

Answer 16

Even the book is saying that they are component vectors.
but termed as scalar for representation:)
u see the attachment carefully.

Answer 17

now I think u must gt it:)

Answer 18

It's kinnnnda both. They're scalars if you know what direction they're pointing in. :P So, if I ask you, what is the x-component of this vector, you could tell me the magnitude of the x-component or the magnitude and direction...but either way there would be no ambiguity since you know what direction was going to be before you got the answer. It's a little more subtle though than magnitude, since this scalar can be negative to indicate pointing in the opposite direction from the corresponding unit vector.

Answer 19

A point in three dimension can be represented \[P(x_1, x_1, x_3)=x_1\cdot\hat {\bf{i}}+x_2\cdot\hat {\bf j} +x_3\cdot\hat {\bf k}\]
\(\{x_1, x_1, x_3\cdots \}\) are the component vectors, which have no direction only magnitude

\(\{\hat {\bf i}, \hat {\bf j}, \hat{ \bf {k}}\}\) are the unit vectors in the three basis dimensions these provide the direction

Answer 20

Again, it's important to note that they have more than just magnitude, they have signed magnitude. It's also okay to represent them as \(\{x_1 \hat \imath,\ x_2 \hat \jmath,\ x_3 \hat k\}\).

Answer 21

@yakeyglee what do you mean when you say "signed magnitude"?

Answer 22

For instance, the vector \(\vec v = 6 \hat \imath - 2 \hat \jmath + 3 \hat k\) has components \(( 6 \hat \imath, - 2 \hat \jmath, 3 \hat k )\). If we were to omit the directions, we would be correct in saying \((6,-2,3)\), however incorrect in simply stating the magnitudes \((6,2,3)\), as that describes a different vector.

Answer 23

They're signed in concordance with their direction with respect to the corresponding unit vectors.

Answer 24

So, sign plays an important role here.
Even if direction is omitted in case of component vector we should not change the sign which were originally assigned to them.
Do i get it right?

Answer 25

Exactly.

Answer 26

thanks @UnkleRhaukus and @yakeyglee ...

Answer 27

I entirely agree with @yakeyglee about the signed scalar.

Actually, I do not think it is no important debate whether vector components are vectors or signed-scalars. It is purely a matter of choice of vocabulary.

In France, we use (or we should use) :
components = vectors
coordinates = scalars (signed of course)

But lots of people interchange them.

We usually say coordinates for vector displacement.