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## The Importance of Vectors ### Physics 11 Advance Placement #### What is a vector? A vector is a mathematical object that has both a magnitude and a direction. Vectors are used to represent physical quantities like displacement, velocity, acceleration, force, and momentum. #### Why are vectors...

## The Importance of Vectors ### Physics 11 Advance Placement #### What is a vector? A vector is a mathematical object that has both a magnitude and a direction. Vectors are used to represent physical quantities like displacement, velocity, acceleration, force, and momentum. #### Why are vectors important? Vectors are important because they allow us to represent and manipulate physical quantities that have both magnitude and direction. This is essential for understanding and predicting the motion of objects, the forces that act on them, and many other physical phenomena. #### How are vectors represented? Vectors can be represented graphically as arrows, where the length of the arrow represents the magnitude of the vector and the direction of the arrow represents the direction of the vector. Vectors can also be represented mathematically using components. For example, a vector in two dimensions can be represented by its $x$ and $y$ components. #### Vector Operations Vectors can be added, subtracted, and multiplied by scalars. ##### Vector Addition To add two vectors, we add their corresponding components. For example, if $\vec{A} = (A_x, A_y)$ and $\vec{B} = (B_x, B_y)$, then $\vec{A} + \vec{B} = (A_x + B_x, A_y + B_y)$. Graphically, vector addition can be represented by placing the tail of one vector at the head of the other vector. The resultant vector is the vector that extends from the tail of the first vector to the head of the second vector. ##### Vector Subtraction To subtract two vectors, we subtract their corresponding components. For example, if $\vec{A} = (A_x, A_y)$ and $\vec{B} = (B_x, B_y)$, then $\vec{A} - \vec{B} = (A_x - B_x, A_y - B_y)$. Graphically, vector subtraction can be represented by adding the negative of the second vector to the first vector. ##### Scalar Multiplication To multiply a vector by a scalar, we multiply each component of the vector by the scalar. For example, if $\vec{A} = (A_x, A_y)$ and $c$ is a scalar, then $c\vec{A} = (cA_x, cA_y)$. Graphically, scalar multiplication changes the magnitude of the vector but not its direction (unless the scalar is negative, in which case the direction is reversed). #### Example Problem A car travels 20.0 km due north and then 35.0 km in a direction $60.0^\circ$ north of east. Find the magnitude and direction of the car’s resultant displacement. **Solution:** First, we break down the second vector into its components: $A_x = (35.0 \, \text{km}) \cdot \cos(60.0^\circ) = 17.5 \, \text{km}$ $A_y = (35.0 \, \text{km}) \cdot \sin(60.0^\circ) = 30.3 \, \text{km}$ Now we can add the components together | Vector | $x$ | $y$ | | :------ | :------- | :------- | | $A_x$ | 17.5 km | 30.3 km | | $B_x$ | 0 km | 20.0 km | | $R_x$ | 17.5 km | 50.3 km | Now we can use Pythagorean theorem to solve for the magnitude of the resultant vector. $c^2 = a^2 + b^2$ $c = \sqrt{a^2 + b^2}$ $R = \sqrt{(17.5 \, \text{km})^2 + (50.3 \, \text{km})^2} = 53.2 \, \text{km}$ $\theta = \tan^{-1} (\frac{50.3}{17.5}) = 70.8^\circ$