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Wize University Calculus 3 Textbook > Vectors & Geometry of Space

Properties of Vectors, Length of Vectors, and Unit Vectors

Vectors in 2-dimensional (2D) and 3-dimensional (3D) SystemsPrevious Section
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Properties of Vectors, Length of Vectors, and Unit Vectors


Given a⃗= <a1, a2, a3>\vec{a}=~<a_1,~a_2,~a_3> and b⃗= <b1, b2, b3>\vec{b}=~<b_1,~b_2,~b_3>, then:

  • a⃗+b⃗= <a1+b1, a2+b2, a3+b3>\vec{a}+\vec{b}=~<a_1+b_1, ~a_2+b_2,~a_3+b_3>
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  • a⃗−b⃗= <a1−b1, a2−b2, a3−b3>\vec{a}-\vec{b}=~<a_1-b_1,~a_2-b_2,~a_3-b_3>
Or we can also look at vector subtraction as the addition of a positive vector a⃗\vec{a} and a negative vector b⃗\vec{b}.

  • <ca1, ca2, ca3>;    c∈ℜ<ca_1,~ca_2,~ca_3>;~~~~c\in\Re
  • So, we multiply all the components by a scalar c∈ℜc\in\Re

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Wize Tip
If a⃗\vec{a} and b⃗\vec{b} are two parallel vectors, then a⃗=cb⃗;c∈ℜ\vec{a}=c\vec{b}; c\in\Re

Properties

  • If v⃗,w⃗,u⃗\vec{v}, \vec{w}, \vec{u} are vectors with an equal number of components and a,b∈ℜa, b \in\Re, then:

1.v⃗+w⃗=w⃗+v⃗2.u⃗+(v⃗+w⃗)=(u⃗+v⃗)+w⃗3.v⃗+0⃗=v⃗4.1v⃗=v⃗5.a(v⃗+w⃗)=av⃗+aw⃗6.(a+b)v⃗=av⃗+bv⃗7.vw⃗=−wv⃗\begin{array}{lrcl} 1.&\vec{v}+\vec{w}&=&\vec{w}+\vec{v}\\\\ 2.&\vec{u}+(\vec{v}+\vec{w})&=&(\vec{u}+\vec{v})+\vec{w}\\\\ 3.&\vec{v}+\vec{0}&=&\vec{v}\\\\ 4.& 1\vec{v}&=&\vec{v}\\\\ 5.& a(\vec{v}+\vec{w})&=&a\vec{v}+a\vec{w}\\\\ 6.& (a+b)\vec{v}&=&a\vec{v}+b\vec{v}\\\\ 7.& \vec{vw}&=&\vec{-wv} \end{array}


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  • Given v⃗=<a1,a2>\vec{v}=<a_1, a_2>, the magnitude (or length/distance) of v⃗\vec{v} in 2-D space is equal to:
∣v⃗∣=a12+a22\boxed{|\vec v|=\sqrt{a_1^2+a_2^2}}
  • Given v⃗=<a1,a2,a3>\vec{v}=<a_1, a_2, a_3> the magnitude (or length/distance) of v⃗\vec{v} in 3-D space is equal to:
∣v⃗∣=a12+a22+a32\boxed{|\vec v|=\sqrt{a_1^2+a_2^2+a_3^2}}
  • Given a vector v⃗\vec{v}, then:
u⃗=v⃗∣v⃗∣\boxed{\vec{u}=\frac{\vec{v}}{|\vec{v}|}}
will be a unit vector that points in the same direction as v⃗\vec{v}.


Remember a unit vector has a magnitude (length) of 1.
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Example

Are vectors m⃗=<13,4,14>\vec{m}=<\frac{1}{3}, 4, \frac{1}{4}> and n⃗=<310,185,940>\vec{n}=<\frac{3}{10}, \frac{18}{5}, \frac{9}{40}> parallel?

In order for two vectors to be parallel, then one vector must be a multiple of the other vector (i.e. a⃗=cb⃗; c∈ℜ\vec{a}=c\vec{b};~c\in\Re).

cm⃗=n⃗c ⋅<13, 4, 14>=<310,185,940>\begin{array}{rcl} c\vec{m}&=&\vec{n}\\\\ c~\cdot<\frac{1}{3},~4,~\frac{1}{4}>&=&<\frac{3}{10}, \frac{18}{5}, \frac{9}{40}> \end{array}

13c=310     →     c=910\frac{1}{3}c=\frac{3}{10}~~~~~{\color{red}{\rightarrow}}~~~~~c=\frac{9}{10}

Verify this fact for the other two components:
4×910 = 185     ✓4\times\frac{9}{10}~=~\frac{18}{5}~~~~~{\color{green}{\checkmark}}

14×910 = 940     ✓\frac{1}{4}\times\frac{9}{10}~=~\frac{9}{40}~~~~~{\color{green}{\checkmark}}

Therefore, vectors m⃗ \vec{m}~and n⃗ \vec{n}~are parallel.
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Example

Find the unit vector that points in the same direction as z⃗=<7,9,−3>\vec{z}=<7, 9, -3>.

Let u⃗ \vec{u}~be the unit vector that points in the same direction as z⃗\vec{z}.

Then,
u⃗=z⃗∣z⃗∣u⃗=<7, 9, −3>72+92+(−3)2\begin{array}{rcl} \vec{u}&=&\dfrac{\vec{z}}{|\vec{z}|}\\\\\\ \vec{u}&=&\dfrac{<7,~9,~-3>}{\sqrt{7^2+9^2+(-3)^2}} \end{array}

u⃗=<7, 9, −3>139\vec{u}=\frac{<7,~9,~-3>}{\sqrt{139}}
Therefore,
u⃗= <7139, 9139, −3139>\vec{u}=~\Bigg<\frac{7}{\sqrt{139}},~\frac{9}{\sqrt{139}},~\frac{-3}{\sqrt{139}}\Bigg>

Practice

Suppose P0=(x0,y0,z0), P_0=(x_0, y_0, z_0),~ where x0, y0, z0 x_0,~y_0,~z_0~ are the components of P0P_0, and P=(16,−3,5/3). P=(16, -3, 5/3).~
If P0P⃗= <19,−3,8/3>\vec{P_0P}=~<19, -3, 8/3>, what are the components of P0⃗?\vec{P_0}?


Practice

Find the unit vector that points in the same direction as a⃗ =<−10, 4, −3>\vec{a}~=<-10,~4,~-3>.