Multiplying vectors -- where does cos(theta) go?

[gracy]

For example if I have to find out work done.Force and displacement are given to be (2i^+3j^+4k^)and
(-i^-2j^) respectively.
till now i have been thinking that work done=Force multiplied by displacement cos theta but in my book in solution of the problem above work done is calculated by simply multiplying
(2i^+3j^+4k^)and
(-i^-2j^).
where did cos theta go?when vectors are given in terms of i^,j^, k^ we don't write costheta,that's how it works,right?if yes,reason? I know displacement cos theta implies component of displacement along force.

 

[Titan97]

That is not a simple multiplication. Its called dot product. Work is defined as:
$$W=\vec{F}\cdot\vec{s}$$

When force and displacement are given in vector forms, you can also find the angle between them by using$$\cos\theta=\frac{\vec{F}\cdot\vec{s}}{|F|.|s|}$$

You can read it here : https://en.m.wikipedia.org/wiki/Dot_product

 

[gracy]

That is not a simple multiplication. Its called dot product. Work is defined as:

W=F⃗ ⋅s⃗​

I know that!but should not magnitude of work done be F S COS theta

 

[Titan97]

Its is.
$$W=|F|.|s|.\cos\theta$$
But you have to find cosθ.
This will be the long way to calculate work though.

 

[Vagn]

For example if I have to find out work done.Force and displacement are given to be (2i^+3j^+4k^)and
(-i^-2j^) respectively.
till now i have been thinking that work done=Force multiplied by displacement cos theta but in my book in solution of the problem above work done is calculated by simply multiplying
(2i^+3j^+4k^)and
(-i^-2j^).
where did cos theta go?when vectors are given in terms of i^,j^, k^ we don't write costheta,that's how it works,right?if yes,reason? I know displacement cos theta implies component of displacement along force.

Your unit vectors i, j, k are mutually orthogonal to one another (at right angles), so cos(θ) is zero for any i≠j terms. You need to look up the dot product in component form for the question above.

 

[gracy]

Your unit vectors i, j, k are mutually orthogonal to one another (at right angles), so cos(θ) is zero for any i≠j terms

And what about i ,-i terms?will it be cos 180 i.e -1?

 

[LittleMrsMonkey]

And what about i ,-i terms?will it be cos 180 i.e -1?

Yes,that's correct.

 

[jbriggs444]

And what about i ,-i terms?will it be cos 180 i.e -1?

The -i component can be understood as (value -1, direction i). With that understanding, when you multiply 2i^ by -i^ the relevant angle is the angle between i^ and i^. That is zero degrees. The result is 2 * -1 * cos(0).

Understood that way, 180 degrees never enters in. All the cosines are equal to 1.

 

[Mark44]

Assuming that both force and displacement are vector constants (i.e., the force doesn't vary and acts along a straight line path), the work done is given by this dot product:
$$W = \vec{F} \cdot \vec{S}$$
Let's assume that $\vec{F} = <f_1, f_2, f_3>$ and $\vec{S} = < s_1, s_2, s_3>$. Personally, I like this notation better than $f_1 \vec{i} + f_2 \vec{j} + f_3 \vec{k}$ as it is easier to write.

There are two ways to calculate the dot product shown above.
Coordinate form
$W = f_1 s_1 + f_2 s_2 + f_3 s_3$

Coordinate-free form
$W = |\vec{F}| |\vec{S}| cos(\theta)$, where $\theta$ is the angle between the two vectors, and |F| and |S| are the magnitudes of the two vectors. For vectors in R³, the magnitude is the square root of the sum of the squares of the three components. For example, $|\vec{F} = \sqrt{f_1^2 + f_2^2 + f_3^2}$.

 

[Mark44]

The -i component can be understood as (value -1, direction i).

Just as valid would be the interpretation that -i has a value of 1 in the direction of -i (i.e., to the left).

With that understanding, when you multiply 2i^ by -i^ the relevant angle is the angle between i^ and i^. That is zero degrees. The result is 2 * -1 * cos(0).

Understood that way, 180 degrees never enters in. All the cosines are equal to 1.

Using my interpretation, we get the same answer via a different route.
$$\vec{2i} \cdot -\vec{i} = |2\vec{i}| |-\vec{i}| \cos(180°) = (2)(1)(-1) = -2$$