58 lines
2.9 KiB
Markdown
58 lines
2.9 KiB
Markdown
A vector is a mathematical concept that denotes direction and magnitude. They're often notated using an arrow ($\vec{v}$), or with a bold, lowercase letter. (**v**).
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Vectors are often denoted as a matrix with two rows: $\begin{bmatrix}1 \\2\end{bmatrix}$
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# Component Form
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If $\vec{v}$ is a vector with the initial point $(x_y,\ y_i)$, and a terminal point $(x_t,\ y_t)$, we can express $\vec{v}$ in component form as $\vec{v} = \langle x_t - x_i,\ y_t, -y_i \rangle$
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# Magnitude
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The magnitude of a vector is $|\vec{v}| = \sqrt{a^2 + b^2}$
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# Direction
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The direction of a vector is $\theta = \tan^-1(\frac{b}{a})$.
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# Addition
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To find $\vec{u} + \vec{v}$, we can put one vector on the end of another vector. The resulting vector will share the same tail as the first vector, and the same head as the second vector.
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# Scalar Multiplication
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A **scalar** is just a real number. Scalar multiplication is multiplying a vector with a real number. This will scale or shrink a vector, but does not change the direction it points at.
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We do not multiply two vectors together.
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# Unit Vector
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A vector with a magnitude of 1 is a **unit vector**.
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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.
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# $i$, $j$ Notation
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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.
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# Trigonometric Form
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Given a vector $\vec{v}$ with a magnitude $|\vec{v}|$ and direction $\theta$:
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The component form is given as:
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$$ \vec{v} = \langle \cos \theta,\ |\vec{v}|\sin\theta \rangle $$
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# Standard position
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- A vector is in standard position if the initial point is at $(0, 0)$.
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# The Dot Product
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The dot product of two vectors $\vec{u} = \langle a, b \rangle$ and $\vec{v} = \langle c, d \rangle$ is $\vec{u} * \vec{v} = ac + bd$.
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- Given that $\vec{u} = \langle -7, 3 \rangle$, and $\vec{v} = \langle -3, 4 \rangle$, find $\vec{u} \cdot \vec{v}$.
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- $\vec{u} \cdot \vec{v} = -7 \cdot -4 + 3 \cdot 4$
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The dot product can be used to find the angle between two vectors.
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If $\theta (0\degree < \theta < 180\degree)$, is the angle between two nonzero vectors $\vec{u}$ and $\vec{v}$, then
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$$ \cos\theta = \dfrac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}|} $$
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# Work
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The dot product can be used to compute the work required to move an object a certain distance.
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To compute work, you need a force and direction. If the force is applied in the same direction:
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$$ W = Fd $$
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The work $W$ is done by a constant force $\vec{F}$ in moving an object from a point $P$ to a point $Q$ is defined by:
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$$ W = \vec{F} \cdot\vec{PQ} = |\vec{F}||\vec{PQ}|\cos\theta $$Where $\theta$ is the angle between $\vec{F}$ and $\vec{PQ}$.
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