Key Concepts

• The position vector has its initial point at the origin.
• If the position vector is the same for two vectors, they are equal.
• Vectors are defined by their magnitude and direction.
• If two vectors have the same magnitude and direction, they are equal.
• Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements.
• Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same.
• Vectors are comprised of two components: the horizontal component along the positive x-axis, and the vertical component along the positive y-axis.
• The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.
• The magnitude of a vector in the rectangular coordinate system is $|v|=\sqrt{{a}^{2}+{b}^{2}}$.
• In the rectangular coordinate system, unit vectors may be represented in terms of $i$ and $j$ where $i$ represents the horizontal component and $j$ represents the vertical component. Then, v = ai + bj  is a scalar multiple of $v$ by real numbers $a\text{ and }b$.
• Adding and subtracting vectors in terms of i and j consists of adding or subtracting corresponding coefficients of i and corresponding coefficients of j.
• A vector v = ai + bj is written in terms of magnitude and direction as $v=|v|\cos \theta i+|v|\sin \theta j$.
• The dot product of two vectors is the product of the $i$ terms plus the product of the $j$ terms.
• We can use the dot product to find the angle between two vectors.
• Dot products are useful for many types of physics applications.

Glossary

dot product

given two vectors, the sum of the product of the horizontal components and the product of the vertical components

initial point

the origin of a vector

magnitude

the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem

resultant

a vector that results from addition or subtraction of two vectors, or from scalar multiplication

scalar

a quantity associated with magnitude but not direction; a constant

scalar multiplication

the product of a constant and each component of a vector

standard position

the placement of a vector with the initial point at $\left(0,0\right)$ and the terminal point $\left(a,b\right)$, represented by the change in the x-coordinates and the change in the y-coordinates of the original vector

terminal point

the end point of a vector, usually represented by an arrow indicating its direction

unit vector

a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the x-axis and is defined as ${v}_{1}=\langle 1,0\rangle$ the vertical unit vector runs along the y-axis and is defined as ${v}_{2}=\langle 0,1\rangle$.

vector

a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)

vector addition

the sum of two vectors, found by adding corresponding components