Vector

Adding vectors on paper using the head to tail method

The Head to Tail method of adding vectors is useful for doing an estimate on paper of the result of adding two vectors. To do it:

• Each vector is drawn as an arrow with an amount of length behind it, where each unit of length on the paper represents a certain magnitude of the vector.
• Draw the next vector, with the tail(end) of the second vector at the head(front) of the first vector.
• Repeat for all further vectors: Draw the tail of the next vector at head of the previous one.
• Draw a line from the tail of the first vector to the head of the last vector - that's the resultant(sum) of all the vectors.

It's called the "Head to Tail" method, because each head from the previous vector leads in to the tail of the next one.

Using component form

^{[clarification needed]}

Using the component form to add two vectors literally means adding the components of the vectors to create a new vector. For example, let a and b be two two-dimensional vectors. These vectors can be written in terms of their components.

${\displaystyle \mathbf {a} =(a_{x},a_{y})}$

${\displaystyle \mathbf {b} =(b_{x},b_{y})}$

Suppose c is the sum of these two vectors, so that c = a + b. This means that ${\displaystyle \mathbf {c} =(a_{x}+b_{x},a_{y}+b_{y})}$.

Here is an example of addition of two vectors, using their component forms:

${\displaystyle \mathbf {a} =(3,-1)}$

${\displaystyle \mathbf {b} =(2,2)}$

${\displaystyle {\begin{aligned}\mathbf {c} &=\mathbf {a} +\mathbf {b} \\&=(a_{x}+b_{x},a_{y}+b_{y})\\&=(3+2,-1+2)\\&=(5,1)\end{aligned}}}$

This method works for all vectors, not just two dimensional ones.

Using the dot product

The dot product is one method to multiply vectors. It produces a scalar. It uses component form:

${\displaystyle {\begin{aligned}\mathbf {a} &=(2,3)\\\mathbf {b} &=(1,4)\\\mathbf {a} \cdot \mathbf {b} &=(2,3)\cdot (1,4)\\&=(2\cdot 1)+(3\cdot 4)\\&=2+12\\&=14\end{aligned}}}$

Using the cross product

The cross product is another method to multiply vectors. Unlike dot product, it produces a vector. Using component form:

${\displaystyle \mathbf {a} \times \mathbf {b} =|\mathbf {a} ||\mathbf {b} |\sin(\theta )\mathbf {n} }$

Here, ${\displaystyle |\mathbf {a} |}$ means the length of ${\displaystyle \mathbf {a} }$, and ${\displaystyle \mathbf {n} }$ is the unit vector at right angles to both ${\displaystyle \mathbf {a} }$ and ${\displaystyle \mathbf {b} }$.

Multiplying by a scalar

To multiply a vector by a scalar (a normal number), you multiply the number by each component of the vector:

${\displaystyle c\,\mathbf {x} =(c\,x_{1},c\,x_{2},...,c\,x_{n})}$

An example of this is

${\displaystyle {\begin{aligned}c&=5\\\mathbf {x} &=(3,4)\\c\,\mathbf {x} &=(5\cdot 3,5\cdot 4)\\&=(15,20)\end{aligned}}}$