# How to Find Vector Components Along Given Directions?

• harshakantha
In summary, this conversation is about finding the components of a vector. The individual products of scalar products are used to solve for the components.
harshakantha
hello! please someone help me, here is my question.

Find the components of d=(3,5,7) along the directions of u, v and w
consider: u=1/3(2,2,-1) v=1/3(2,-1,2) w=1/3(-1,2,2)

I don't know where to start, I need some ideas to solve this
thanx

Start with the definition of "component of a vector". Think of scalar product.

ehild

please tell me how to apply scalar product, I know what is scalar product, but I don't know how to use it for find components

You can write up a vector as a linear combination of the base vectors. a=x1u+x2v+x3w. x1, x2, x3 are the components of vector a. Notice that u,v,w are orthonormal. What do you get if multiplying the equation with one of them?

ehild

I don't understand, with which do I need to multiply?? can you show me how to do that?

Multiply both sides of the equation d=x1u+x2v+x3w by u:
du=x1uu+x2vu+x3wu.
ehild

How do I suppose to calculate du? by using the scalar product??

after solving du=x1uu+x2vu+x3wu, I got 27 for x1 is this correct ehild?

No, how did you get it?

ehild

I used scalar product to solve it,

du=x1uu+x2vu+x3wu
(3,5,7)1/3(2,-2,-1)=x11/9(2,-2,-1)(2,-2,-1)+x21/3(2,-1,2)1/3(2,2,-1)+x31/3(-1,2,2)1/3(2,2,-1)

What did you get for the individual products du, uu, vu, wu?

ehild

I'm sorry, what did you mean by individual products, is it du,x1uu,x2uv and x3uw,if so x2uv and x3uw become zero,

Yes, but how much is uu?

ehild

oops I've made a mistake when solving , is UU=1, then I got 3 for x1

Very good! Now do the same (without mistake) with v and w.

Note: this method works for orthogonal u, v ,w vectors only. In general, you can write a linear system of equations for x1,x2,x3 and solve it.

ehild

Oh...thank you very much ehild, I really appreciate your help, I have another problem, can you explain me little bit about mutually perpendicular unit vectors

You had such ones just now. u, v, w are mutually perpendicular if the scalar product of any two of them is zero, and a vector is unit vector if its modulus (or magnitude) is 1. You get it by multiplying the vector by itself and taking the square root.
You can make an unit vector of any vector by dividing all components by the modulus.

Can I help something more?

ehild

ehild said:
Very good! Now do the same (without mistake) with v and w.

Note: this method works for orthogonal u, v ,w vectors only. In general, you can write a linear system of equations for x1,x2,x3 and solve it.

ehild

I found 3, 5 and 7 respectively for the components x1,x2 and x3. but I have a doubt about these values,

I had to Find the components of d=(3,5,7) along the directions of u, v and w
consider: u=1/3(2,2,-1) v=1/3(2,-1,2) w=1/3(-1,2,2), finally I got 3,5,7. this is really confusing me

Your result means that by decomposing the vector d to three vectors parallel to u,v, w, these component vectors are
3u, 5v and 7 w, that is: d=3u+5v+7w. Check.

ehild

ehild said:
Your result means that by decomposing the vector d to three vectors parallel to u,v, w, these component vectors are
3u, 5v and 7 w, that is: d=3u+5v+7w. Check.

ehild

Thanx ehild, then what is my final answer would be?

It depends how "component" was defined during your classes; I would say 3u, 5v, 7w.

ehild

Thanx a lot ehild bye..

ehild said:
You had such ones just now. u, v, w are mutually perpendicular if the scalar product of any two of them is zero, and a vector is unit vector if its modulus (or magnitude) is 1. You get it by multiplying the vector by itself and taking the square root.
You can make an unit vector of any vector by dividing all components by the modulus.

Can I help something more?

ehild

thanks ehild your post really helpful to me, now I got a more clear idea about mutually perpendicular unit vectors :), bye..

Splendid!

ehild

## 1. What is a vector?

A vector is a mathematical object that has both magnitude and direction. It is represented by an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

## 2. What are the components of a vector?

The components of a vector are the parts that make up the vector in a specific coordinate system. These components are usually represented as x and y for two-dimensional vectors, and x, y, and z for three-dimensional vectors.

## 3. How do you find the magnitude of a vector?

The magnitude of a vector can be found using the Pythagorean theorem, where the magnitude is equal to the square root of the sum of the squared components. In other words, the magnitude is the length of the vector.

## 4. How do you find the direction of a vector?

The direction of a vector can be found using trigonometric functions such as sine, cosine, and tangent. This is usually done by finding the angle between the vector and a reference axis.

## 5. What is the difference between a vector's components and its magnitude and direction?

The components of a vector are the parts that make up the vector in a specific coordinate system, while the magnitude and direction describe the overall properties of the vector. The components are numerical values, while the magnitude is a measurement of length and the direction is a measurement of angle.

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