The document Cartesian Vectors Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Engineering Mechanics - Notes, Videos, MCQs & PPTs.

All you need of Mechanical Engineering at this link: Mechanical Engineering

**Cartesian Force Vectors**

Any force** F** can be written in the form of a Cartesian vector. Such an expression uses the addition of the forceâ€™s component vectors in the *x*, *y*, and *z* directions of the axes of the right-hand coordinate. If the magnitude of its component in the x direction is Fx, in the y direction Fy, an in the z direction Fz, then F is expressed as

**F** = *F _{x}*

where**i**, **j**, and **k** are the Cartesian unit vectors.

**Expressing A Force Vector By Using Its Magnitude And Two Points On Its Line Of Action.** The Cartesian vector expression for a force is often obtained by using the *Cartesian position vector* that gives the Cartesian unit vectors corresponding to the direction concerned. For example, the force **F** has a line of action that passes through points *P(x _{1},y_{1},z_{1})* and

Let the position vector **r** that points from *P* to *Q* describes the relative position between *P* and *Q*. From the diagram, we obtain the relationship

**r** = â€“**r**_{1} +** r**_{2}

However, we can formulate the relationships **r**_{1} = *x*_{1}**i** + *y _{1}*

** r** = (*x _{2}* â€“

Hence the components of **r** in the *x, y,* and *z* directions are obtained by subtracting the coordinates of the tail from those of the heads. Note that if the coordinates of *Q* are subtracted from those of *P*, we will get the vector â€“**r**.

From Equation 2.22, we will get the unit vector **s** (corresponding to the line of action and the direction of **F)** as

s = r

= ** â€¦..(2.3)**

In this expression, as usual, *r* is the magnitude of the position vector **r** and is given by the relationship

*r* =

A force of magnitude *F* and having its line of action, direction, and sense corresponding to the unit vector s in Figure 2.3 can thus be expressed in the Cartesian vector form by writing **F** = *F***s **and then expanding the expression on the right of the equal sign.

The coordinate direction angles for the line of action of **F** are obtained from the cosine direction relationship for **s**, i.e.

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!

30 videos|72 docs|65 tests

### Addition and Subtraction of Cartesian Vectors

- Video | 07:34 min
### Test: Addition Of A System Of Coplanar Forces

- Test | 14 ques | 15 min
### Addition and Subtraction of Cartesian Vectors

- Doc | 1 pages
### Position Vectors

- Doc | 1 pages
### Dot Product of Vectors

- Doc | 3 pages
### Dot Product of Vectors

- Video | 04:08 min

- Test: Cartesian Vectors
- Test | 30 ques | 30 min