 maths > vector-algebra

Properties of Magnitude of a Vector

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Overview

Properties of Magnitude of Vectors

»  Magnitude of null vectors $|\stackrel{\to }{0}|$$| \vec{0} |$$=0$$= 0$

»  Magnitude of a negative $|-\stackrel{\to }{p}|=|\stackrel{\to }{p}|$$| - \vec{p} | = | \vec{p} |$

»  Magnitude of an unit vector $|\stackrel{^}{p}|=1$$| \hat{p} | = 1$

properties

Null vector is $\stackrel{\to }{0}=0i+0j+0k$$\vec{0} = 0 i + 0 j + 0 k$. The magnitude of null vector is $|\stackrel{\to }{0}|=\sqrt{{0}^{2}+{0}^{2}+{0}^{2}}=0$$| \vec{0} | = \sqrt{{0}^{2} + {0}^{2} + {0}^{2}} = 0$.

Magnitude of a Null vector: Null vector is $\stackrel{\to }{0}=0i+0j+0k$$\vec{0} = 0 i + 0 j + 0 k$. The magnitude $|\stackrel{\to }{0}|=0$$| \vec{0} | = 0$

The magnitude of a null vector is zero.

Given a vector $\stackrel{\to }{p}$$\vec{p}$, The magnitude of negative of the vector $=|\stackrel{\to }{p}|$$= | \vec{p} |$

$\stackrel{\to }{p}={p}_{x}i+{p}_{y}j+{p}_{z}k$$\vec{p} = {p}_{x} i + {p}_{y} j + {p}_{z} k$

The negative of $\stackrel{\to }{p}$$\vec{p}$ is $-\stackrel{\to }{p}=-{p}_{x}i-{p}_{y}j-{p}_{z}k$$- \vec{p} = - {p}_{x} i - {p}_{y} j - {p}_{z} k$ and so
$|-\stackrel{\to }{p}|$$| - \vec{p} |$
$\quad \quad = \sqrt{{\left(- {p}_{x}\right)}^{2} + {\left(- {p}_{y}\right)}^{2} + {\left(- {p}_{z}\right)}^{2}}$
$\quad \quad = \sqrt{{\left({p}_{x}\right)}^{2} + {\left({p}_{y}\right)}^{2} + {\left({p}_{z}\right)}^{2}}$
$\quad \quad = | \vec{p} |$

Magnitude of negative vector: For a vector $\stackrel{\to }{p}$$\vec{p}$, the magnitude of the negative $|-\stackrel{\to }{p}|=|\stackrel{\to }{p}|$$| - \vec{p} | = | \vec{p} |$

Magnitude of the negative of a vector equals magnitude of the vector.

The magnitude of a proper vector $\ne 0$$\ne 0$

Given a proper vector $\stackrel{\to }{p}={p}_{x}i+{p}_{y}j+{p}_{z}k$$\vec{p} = {p}_{x} i + {p}_{y} j + {p}_{z} k$,

By definition, a proper vector is not a null vector. That is, at least one of the components ${p}_{x},{p}_{y},{p}_{z}$${p}_{x} , {p}_{y} , {p}_{z}$ is non-zero, and can be a positive or negative value. So the magnitude $\sqrt{{\left({p}_{x}\right)}^{2}+{\left({p}_{y}\right)}^{2}+{\left({p}_{z}\right)}^{2}}$$\sqrt{{\left({p}_{x}\right)}^{2} + {\left({p}_{y}\right)}^{2} + {\left({p}_{z}\right)}^{2}}$ is not zero and positive.

Magnitude of Proper Vector: For a proper vector $\stackrel{\to }{p}$$\vec{p}$, the magnitude $|\stackrel{\to }{p}|\ne 0$$| \vec{p} | \ne 0$ and $|\stackrel{\to }{p}|>0$$| \vec{p} | > 0$

Magnitude of a proper vector is not zero and positive.

The magnitude of a unit vector is $1$$1$

By definition, magnitude of an unit vector is 1.

Magnitude of a unit vector: For a unit vector $\stackrel{\to }{p}$$\vec{p}$, $|\stackrel{\to }{p}|=1$$| \vec{p} | = 1$

Magnitude of an unit vector is 1

summary

»  Magnitude of null vectors $|\stackrel{\to }{0}|$$| \vec{0} |$$=0$$= 0$

»  Magnitude of a negative $|-\stackrel{\to }{p}|=|\stackrel{\to }{p}|$$| - \vec{p} | = | \vec{p} |$

»  Magnitude of an unit vector $|\stackrel{^}{p}|=1$$| \hat{p} | = 1$

Outline