In this page, the parallelogram law of vector addition is explained as a continuous addition of vectors.

*click on the content to continue..*

Note: Using Geometry, the triangle property of vector addition can be extended to explain parallelogram property of vector addition. In the coming pages, we establish a different point of view for the same.

Two taps simultaneously fill a bucket. One fills at `20` liter per hour and another fills at `10` liter per hour. What type of addition is this?

- Sequential addition
- Continuous addition
- Continuous addition

Answer is 'continuous addition'. It is called continuous addition as the quantities are added continuously in small quantities over time. *Scalar quantities can be added continuously.*.

A ball rolls with two velocity components `2`m/sec at `11^@`angle and `1.3`m/sec at `67^@` angle. The effective velocity of the ball is calculated using, which of the following?

- Sequential addition of vectors
- Continuous addition of vectors
- Continuous addition of vectors
- This does not look to be vector addition

answer is 'Continuous addition of vectors' *The two vectors are acting together resulting in a vector sum.*

A table is pulled by two persons with forces `f_a` and `f_b` with an angle `theta` between the two forces. The effective force on the table is calculated using in which of the following?

- Sequential addition of vectors
- Continuous addition of vectors
- Continuous addition of vectors
- This does not look to be vector addition

answer is 'Continuous addition of vectors' * The two vectors are acting together resulting in a vector sum.*

Continuous addition of two vectors is shown in the figure.

The magnitude of a vector is equivalently shown as length of the ray in a coordinate plane. Let us see how to calculate `vec r`.

The vectors `vec x` and `vec y` can be split into components

• `vec x` = `vec a` + `vec b`

• `vec y` = `vec p` + `vec q`

Note that `vec b` = `-vec q`

and `vec a` and `vec p` are in the same direction.

The component vectors are equivalently rearranged as shown in the figure. How will `vec a` and `vec p` interact?

- Add in magnitude
- Add in magnitude
- Subtract in magnitude
- cannot calculate

Answer is 'add in magnitude', as they are in the same direction

The component vectors are equivalently rearranged as shown in the figure. How will `vec b` and `vec q` interact?

- cancel out to `0`
- subtract in magnitude
- subtract in magnitude
- both the above

Answer is 'both the above', as they are in the opposite direction.

Note that the result forms a diagonal to the parallelogram. This is given as the parallelogram property of vector addition.

Two vectors form a parallelogram and the *co-initial diagonal is the sum*.

** Parallelogram Property of vector addition ** when two vectors are added, arrange such that their initial points coincide. In that configuration, complete a parallelogram with the two vectors as the two adjacent sides. The diagonal that starts from the initial point of the vectors is the sum of the two vectors.

*slide-show version coming soon*