Defining Vectors

Vectors are a list of components. They can be expressed in ij notation by:

a = 2 i + 3 j - 4 k

a = 2 i + 3 j - 4 k

You can also express a vector as a matrix:

a = 23 - 4 a = [23 - 4]

Adding and Subtracting Vectors

To add vectors, add their corresponding components. For example:

a = 4 i + 7 j - 9 k b = 3 i - 5 j - 8 k a + b = 7 i + 2 j - 17 k

Subtracting vectors works in a similar fashion:

a - b = i + 12 j - k

Here are the formulas:

a + b = a_{i} + b_{i} a - b = a_{i} - b_{i}

Here’s a graph visualizing the addition and subtraction of vectors: https://www.desmos.com/calculator/gavjpwhnuo

Multiplication by Scalar

To multiply a vector by a scalar (regular number), just multiply all the components by that number:

m a = m a_{i}

Multiplication by Another Vector: Dot Product

There are two different ways to multiply a vector by another vector. The first way is a dot product. Here is the algebraic definition, where n is the length of the two vectors:

a \cdot b = i = 0 \sum n a_{i} b_{i}

With two two dimensional vectors, we can also provide a geometric definition, where $∣∣ a ∣∣$ is the magnitude of $a$ , and $θ$ is the angle between the vectors:

a \cdot b = ∣∣ a ∣∣ ∣∣ b ∣∣ cos θ

As you can see, the dot product returns a single value, or scalar. From the geometric definition, you can see that it describes how much one vector “aligns” to the other.

Proving that the Definitions are the Same

Let $a$ have a magnitude of $m$ and an angle of $p$ , let $b$ have a magnitude of $n$ and an angle of $q$ .

a \cdot b = m cos p n cos q + m sin p n sin q = mn (cos p cos q + sin p sin q) = mn cos (p - q)

Using the algebraic definition, we can get the geometric definition as shown above.

Cross Product

Let $n$ be a unit vector perpendicular to $a$ and $b$ , and $θ$ be the angle between them. The cross product is:

a \times b = ∣∣ a ∣∣ ∣∣ b ∣∣ sin θ n

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Vectors