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education/math/MATH1060 (trig)/Vectors.md
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@ -23,14 +23,14 @@ A vector with a magnitude of 1 is a **unit vector**.
If $\vec{v}$ is a nonzero vector, the unit vector can be found using the equation $\vec{u} = \dfrac{1}{|\vec{v}|}\vec{v}$ . In other words, to find a unit vector, divide the vector by its magnitude. If $\vec{v}$ is a nonzero vector, the unit vector can be found using the equation $\vec{u} = \dfrac{1}{|\vec{v}|}\vec{v}$ . In other words, to find a unit vector, divide the vector by its magnitude.
# $i$, $j$ Notation # $i$, $j$ Notation
Every 2d vector has a horizontal component and a vertical component. The horizontal unit vector could be written as $i = \langle 1, 0 \rangle$, and the vertical unit vector could be written as $j = \langle 0, 1 \rangle$ Every vector can be made up using a combination of these standard unit vectors. Every 2D vector has a horizontal component and a vertical component. The horizontal unit vector could be written as $i = \langle 1, 0 \rangle$, and the vertical unit vector could be written as $j = \langle 0, 1 \rangle$ Every vector can be made up using a combination of these standard unit vectors.
# Trigonometric Form # Trigonometric Form
Given a vector $\vec{v}$ with a magnitude $|\vec{v}|$ and direction $\theta$: Given a vector $\vec{v}$ with a magnitude $|\vec{v}|$ and direction $\theta$:
The component form is given as: The component form is given as:
$$ \vec{v} = \langle \cos \theta,\ |\vec{v}|\sin\theta \rangle $$ $$ \vec{v} = \langle |\vec{v}||\cos \theta,\ |\vec{v}|\sin\theta \rangle $$
# Standard position # Standard position
- A vector is in standard position if the initial point is at $(0, 0)$. - A vector is in standard position if the initial point is at $(0, 0)$.
@ -38,13 +38,13 @@ $$ \vec{v} = \langle \cos \theta,\ |\vec{v}|\sin\theta \rangle $$
# The Dot Product # The Dot Product
The dot product of two vectors $\vec{u} = \langle a, b \rangle$ and $\vec{v} = \langle c, d \rangle$ is $\vec{u} * \vec{v} = ac + bd$. The dot product of two vectors $\vec{u} = \langle a, b \rangle$ and $\vec{v} = \langle c, d \rangle$ is $\vec{u} * \vec{v} = ac + bd$.
- Given that $\vec{u} = \langle -7, 3 \rangle$, and $\vec{v} = \langle -3, 4 \rangle$, find $\vec{u} * \vec{v}$. - Given that $\vec{u} = \langle -7, 3 \rangle$, and $\vec{v} = \langle -3, 4 \rangle$, find $\vec{u} \cdot \vec{v}$.
- $\vec{u} * \vec{v} = -7 * -4 + 3 * 4$ - $\vec{u} \cdot \vec{v} = -7 \cdot -4 + 3 \cdot 4$
The dot product can be used to find the angle between two vectors. The dot product can be used to find the angle between two vectors.
If $\theta (0\degree < \theta < 180\degree)$, is the angle between two nonzero vectors $\vec{u}$ and $\vec{v}$, then If $\theta (0\degree < \theta < 180\degree)$, is the angle between two nonzero vectors $\vec{u}$ and $\vec{v}$, then
$$ \cos\theta = \dfrac{\vec{u}*\vec{v}}{|\vec{u}||\vec{v}|} $$ $$ \cos\theta = \dfrac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|} $$
# Work # Work
The dot product can be used to compute the work required to move an object a certain distance. The dot product can be used to compute the work required to move an object a certain distance.