Section 1.2 Vector Operations
Definition 1.2.1.
Let \(\vect{v}=\ot{v_1}{v_2}{v_3}\) and \(t \in \mathbb{R}\text{.}\) The scalar product of \(t\) and \(\vect{v}\) is defined by: \(t\vect{v} = \ot{tv_1}{tv_2}{tv_3}\)
Definition 1.2.2.
Let \(\vect{v}=\ot{v_1}{v_2}{v_3}\) and \(\vect{w}=\ot{w_1}{w_2}{w_3}\text{.}\) The sum of \(\vect{v}\) and \(\vect{w}\) is defined by: \(\vect{v}+\vect{w} =\ot{v_1+w_1}{v_2+w_2}{v_3+w_3}\text{.}\)
These are algebraic definition. It is also important to understand how these definitions affect vectors geometrically.
Checkpoint 1.2.3.
Use the vector \(\vect{v}\) below and give geometric representations of the vectors described by the scalar products: \(3\vect{v}\) and \(-1\vect{v}\text{.}\)
Checkpoint 1.2.4.
Given \(\vect{v} = \ot{1}{-2}{2}\text{,}\) find a vector with the same direction as \(\vect{v}\text{,}\) but with length \(1\text{,}\) with length \(10\text{,}\) and with length \(h\text{.}\)
Checkpoint 1.2.5.
Use the vectors \(\vect{v}\) and \(\vect{w}\) below to give a geometric representation of the vector which is the sum \(\vect{v}+\vect{w}\text{.}\) If this interpretation is applied to \(\vect{w}\)+\(\vect{v}\) is the result the same? If this interpretation is applied to \((\vect{v} +\vect{w}) + \vect{u}\) and \(\vect{v} + ( \vect{w} + \vect{u})\) are those results the same?
Theorem 1.2.6.
For any ordered triple \(\vect{v}=\ot{v_1}{v_2}{v_3} \in \mathbb{R}^{3}\) there exists an ordered triple \(\vect{x}=\ot{x_1}{x_2}{x_3} \in \mathbb{R}^{3}\) such that when \(\vect{x}\) is added to \(\vect{v}\) the result is the origin, \(\vect{0}=\ot{0}{0}{0}\text{.}\) For a given \(\vect{v}\) such an ordered triple is unique. Geometrically the vector \(\vect{x}\) can be visualized as . (Describe starting and ending positions, or direction and length.)
Also, for any ordered triples \(\vect{v}=\ot{v_1}{v_2}{v_3} \in \mathbb{R}^{3}\) and \(\vect{w}=\ot{w_1}{w_2}{w_3} \in \mathbb{R}^{3}\) there exists an ordered triple \(\vect{y}=\ot{y_1}{y_2}{y_3} \in \mathbb{R}^{3}\) such that when \(\vect{y}\) is added to \(\vect{v}\) the result is \(\vect{w}=\ot{w_1}{w_2}{w_3}\text{.}\) For any given \(\vect{v}\) and \(\vect{w}\) such an ordered triple is unique. Geometrically the vector \(\vect{y}\) can be visualized as . (Describe starting and ending positions, or direction and length.)
Theorem 1.2.6 provides justification for the following definitions.
Definition 1.2.7.
Let \(\vect{v}=\ot{v_1}{v_2}{v_3} \in \mathbb{R}^3\text{.}\) The additive inverse of \(\vect{v}\) is the ordered triple, written \(-\vect{v}\text{,}\) that, when added to \(\vect{v}\) results in the origin: \(\vect{0} = \ot{0}{0}{0}\text{.}\) In symbols \(-\vect{v}\) is the ordered triple such that \(\vect{v} + -\vect{v}=0\text{.}\)
Definition 1.2.8.
Let \(\vect{v}=\ot{v_1}{v_2}{v_3} \in \mathbb{R}^3\) and \(\vect{w}=\ot{w_1}{w_2}{w_3} \in \mathbb{R}^3\text{.}\) The difference of \(\vect{w}\) from \(\vect{v}\) is the ordered triple, written \(\vect{w}-\vect{v}\text{,}\) which when added to \(\vect{v}\) results in \(\vect{w}\text{.}\) In symbols \(\vect{w}-\vect{v}\) is the ordered triple such that \(\vect{v} +(\vect{w}-\vect{v})=\vect{w}\text{.}\)
Discussion Question 1.2.9.
Do \(\vect{w}-\vect{v}\) and \(\vect{v}-\vect{w}\) describe the same ordered triple? Do \(\vect{w}-\vect{v}\) and \(\vect{w} + \vect{-v}\) describe the same ordered triple?
Challenge 1.2.10.
Let \(\vect{v}=\ot{v_1}{v_2}{v_3} \in \mathbb{R}^3\) and \(k \in \mathbb{R}\text{.}\) Does \(\|k\vect{v}\| = k\|\vect{v}\|\text{?}\) Does \(\| \vect{v} + \vect{w}\| =\|\vect{v}\|+\|\vect{w}\|\text{?}\) Include justification.