4.2 Vectors

Vectors in space are the three \(-\) dimensional analog of vectors in the plane and are subject to the same rules of addition, subtraction and scalar multiplication that govern vectors in the plane. The vectors from the origin to the points \((1,0,0) , (0,1,0)\) and \((0,0,1)\) are the basic unit vectors. We denote them by \(\textbf {i}, \textbf {j}\) and \(\textbf {k}\), respectively. We write the vectors from the origin \(O\) to the point \(P(x,y,z)\) as \(\overline {v} = \overrightarrow {OP} = x\textbf {i} + y\textbf {j} + z\textbf {k}\). If \(P_1(x_1,y_1,z_1)\) and \(P_2(x_2,y_2,z_2)\) are two points then the vectors from \(P_1\) to \(P_2\) is the vector \[\overrightarrow {P_1P_2} = (x_2 - x_1)\textbf {i} + (y_2 - y_1)\textbf {j} + (z_2 - z_1)\textbf {k}\]

The length (magnitude) of any vector \(A = a\textbf {i} + b\textbf {j} + c\textbf {k}\) is given by \[\left |A\right | = \sqrt {a^2 + b^2 + c^2}\]

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