9Avectorops


 * Operations with Vectors**

Arithmetic with vectors is very similar to arithmetic with complex numbers.

Null Vector

A vector with a magnitude of __zero__ is called the ** Null Vector **.

Multiply by a Scalar If a vector is multiplied by a scalar (an ordinary number) then the vector maintains the same direction and changes magnitude.

Example

math \\ \overrightarrow{AB}=2\underline{i}+4\underline{j} \qquad. \\ . \\ 3\overrightarrow{AB} = 3(2\underline{i}+4\underline{j})=6\underline{i}+12\underline{j} \qquad. math

In matrix notation, this would be: math . \qquad 3 \left[ \begin{matrix} 2&4 \end{matrix} \right] = \left[ \begin{matrix} 6&12 \end{matrix} \right] \qquad. math

Remember: The calculator uses matrix notation for vectors, but you must use __**i**__ and __**j**__ notation.

Addition

If two vectors are added together, the head (arrow end) of the first vector is joined to the tail (back end) of the second vector. The resultant vector starts at the tail of the first vector and goes to the head of the second vector. Example

math \overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC} \qquad. math

Using **__i__** and **__j__** notation, add the **__i__** components together and add the **__j__** components together.

math \big( 2\underline{i}+4\underline{j} \big) + \big( 3\underline{i}+8\underline{j} \big) = 5\underline{i}+12\underline{j} \qquad. math

In matrix notation, this would be: math .\qquad \left[ \begin{matrix} 2&4 \end{matrix} \right] + \left[ \begin{matrix} 3&8 \end{matrix} \right] = \left[ \begin{matrix} 5&12 \end{matrix} \right] math

The Negative Vector The negative of a vector has the same magnitude and the reverse direction

Example

math \\ -\overrightarrow{AB}=\overrightarrow{BA} \qquad \\. \\ -\big( 2\underline{i}+4\underline{j} \big) = -2\underline{i}-4\underline{j} \qquad. math

math \\ \overrightarrow{AB} + \overrightarrow{BA} = \textit{ null vector} \qquad. \\ . \\ \big( 2\underline{i}+4\underline{j} \big) + \big( -2\underline{i}-4\underline{j} \big) = 0\underline{i} + 0\underline{j} \qquad. math

Subtraction

To subtract, take the negative of the second vector and add.

Example math \\ \overrightarrow{AB}-\overrightarrow{BC}=\overrightarrow{AB}+\overrightarrow{CB} \qquad. \\ . \\ . \qquad \qquad =\overrightarrow{AB}+\overrightarrow{BD} \qquad. \\ . \\ . \qquad \qquad =\overrightarrow{AD} math

Using **__i__** and **__j__** notation, subtract the **__i__** components and subtract the **__j__** components.

math \big( 2\underline{i}+4\underline{j} \big) - \big( 3\underline{i}+8\underline{j} \big) = -1\underline{i}-4\underline{j} \qquad. math

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