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Vectors

Unit vectors

Unit vectors have a magnitude of 1.

The basic unit vectors are i = \left( \matrix{ 1 \cr 0 \cr 0 \cr} \right), j = \left( \matrix{ 0 \cr 1 \cr 0 \cr} \right) and k = \left( \matrix{ 0 \cr 0 \cr 1 \cr} \right)

Any vector can be written in terms of i, j and k. For example,

\eqalign{ \left( \matrix{ \;\;3 \cr \;\;4 \cr - 2 \cr} \right)\; & = & \;\left( \matrix{ 3 \cr 0 \cr 0 \cr} \right)\; + \;\left( \matrix{ 0 \cr 4 \cr 0 \cr} \right)\; + \;\left( \matrix{ \;\;0 \cr \;\;0 \cr - 2 \cr} \right)\; \cr \cr \cr & = & \;3\left( \matrix{ 1 \cr 0 \cr 0 \cr} \right)\; + \;4\left( \matrix{ 0 \cr 1 \cr 0 \cr} \right)\; - \;2\left( \matrix{ 0 \cr 0 \cr 1 \cr} \right)\; \cr\cr & = & \;3i\; + 4j\; - \;2k \cr}

Question

Express \left( \matrix{ \;\;5 \cr \;\;0 \cr - 1 \cr} \right) in terms of unit vectors.

Answer

\left( \matrix{ \;\;5 \cr \;\;0 \cr - 1 \cr} \right) = 5i + 0j - 1k

5i +0j -1k = 5i - k

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