Vector Addition

Adding Displacements Tip to Tail

Vector addition differs from scalar addition. In scalar addition, we only add magnitudes without considering direction. For example, $2 \text{ kg}$ of sugar plus $3 \text{ kg}$ of sugar equals $5 \text{ kg}$ of sugar. However, in vector addition, we must consider both magnitude and direction.

For instance, if someone walks $2 \text{ m}$ east and then $3 \text{ m}$ west, the result differs from walking $2 \text{ m}$ east and then $3 \text{ m}$ east again. The result of vector addition is called the resultant vector.

Vector Addition Using the Triangle Method

1. Draw the first vector $\vec{a}$
2. Translate the second vector $\vec{b}$ until its tail is at the tip of the first vector
3. Draw the resultant from the tail of the first vector to the tip of the second

Mathematically, if $\overrightarrow{AB}$ and $\overrightarrow{BC}$ are two vectors, then:

Vector addition satisfies the commutative property:

Vector Addition Using the Parallelogram Method

Another method for adding vectors is the parallelogram method. The steps are:

1. Draw both vectors with their tails coinciding
2. Create a parallelogram using the two vectors as sides
3. The resultant vector is the diagonal of the parallelogram passing through the common origin of both vectors

This method also satisfies the commutative property, so the order of addition does not affect the result.

Addition Using the Polygon Method

To add more than two vectors, we can use the polygon method. The principle is that the tip of the previous vector meets the tail of the next vector. The resultant vector connects the tail of the first vector to the tip of the last vector.

Vector addition using the polygon method also satisfies the associative property:

To add three vectors using the parallelogram method, we can:

1. Add two vectors first to get the resultant $R_1$
2. Add $R_1$ to the third vector to get the final resultant $R_2$

In the following two diagrams, $A=(3,1,0)$, $B=(1,3,0)$, and $C=(0,1,3)$. The intermediate sums are $B+C=(1,4,3)$ and $A+B=(4,4,0)$. Both groupings give $A+B+C=(4,5,3)$.

Vector Addition by Components

Besides graphical methods, vectors can also be added by their components. In a coordinate system, each vector can be expressed in terms of its components.

If $\vec{a} = (a_x, a_y)$ and $\vec{b} = (b_x, b_y)$, then:

Example:

Addition by components gives the same result as graphical addition.

Boat Velocity in a River

Vector addition can determine the velocity of a boat crossing a river:

1. If the river current velocity is zero, the boat's velocity relative to the shore equals its velocity relative to the water
2. If the river current velocity is nonzero, the boat's velocity relative to the shore is the vector sum of its water-relative velocity and the current velocity. Whether its direction, magnitude, or both change depends on the current's direction

A sequence of displacement vectors can likewise be added. Its resultant gives the net displacement. Route length and the shortest route must be determined from the path of the journey.

Difference Between Scalar and Vector Addition

Both scalar addition and vector addition produce a unique result for fixed inputs. The difference is that scalar inputs carry only magnitude, while vector inputs also carry direction. Changing a vector's direction therefore changes the input and can change the resultant.

Example of scalar addition:

For sugar, $3 \text{ kg} + 4 \text{ kg} = 7 \text{ kg}$.

Example of vector addition:

For displacement in the same direction, $3 \text{ m east} + 4 \text{ m east} = 7 \text{ m east}$. In opposite directions, $3 \text{ m east} + 4 \text{ m west} = 1 \text{ m west}$.

Vector addition calculates resultant displacement, velocity, acceleration, and force.