Essential Concepts
- Vectors are used to represent quantities that have both magnitude and direction.
- We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.
- Subtraction of vectors is defined in terms of adding the negative of the vector.
- A vector is written in component form as [latex]{\bf{v}}=\langle{x,y}\rangle[/latex].
- The magnitude of a vector is a scalar: [latex]\parallel{\bf{v}}\parallel=\sqrt{x^2+y^2}[/latex].
- A unit vector [latex]\bf{u}[/latex] has magnitude [latex]1[/latex] and can be found by dividing a vector by its magnitude: [latex]{\bf{u}}=\frac{1}{\parallel{\bf{v}}\parallel}{\bf{v}}[/latex] The standard unit vectors are [latex]{\bf{i}}=\langle{1,0}\rangle[/latex] and [latex]{\bf{j}}=\langle{0,1}\rangle[/latex]. A vector [latex]{\bf{v}}=\langle{x,y}\rangle[/latex] can be expressed in terms of the standard unit vectors as [latex]{\bf{v}}=x{\bf{i}}+y{\bf{j}}[/latex].
- Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.
Glossary
- component
- a scalar that describes either the vertical or horizontal direction of a vector
- equivalent vectors
- vectors that have the same magnitude and the same direction
- initial point
- the starting point of a vector
- magnitude
- the length of a vector
- normalization
- using scalar multiplication to find a unit vector with a given direction
- parallelogram method
- a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram
- scalar
- a real number
- scalar multiplication
- a vector operation that defines the product of a scalar and a vector
- standard unit vectors
- unit vectors along the coordinate axes: [latex]{\bf{i}}=\langle{1,0}\rangle[/latex], [latex]{\bf{j}}=\langle{0,1}\rangle[/latex]
- standard-position Vectors
- a vector with initial point [latex](0,0)[/latex]
- terminal point
- the endpoint of a vector
- triangle inequality
- the length of any side of a triangle is less than the sum of the lengths of the other two sides
- triangle method
- a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector
- unit vector
- a vector with magnitude [latex]1[/latex]
- vector
- a mathematical object that has both magnitude and direction
- vector addition
- a vector operation that defines the sum of two vectors
- vector difference
- the vector difference [latex]{\bf{v}}-{\bf{w}}[/latex] is defined as [latex]{\bf{v}}+(-{\bf{w}})={\bf{v}}+(-1){\bf{w}}[/latex]
- vector sum
- the sum of two vectors, [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex]can be constructed graphically by placing the initial point of [latex]{\bf{w}}[/latex] at the terminal point of [latex]{\bf{v}}[/latex]; then the vector sum [latex]{\bf{v}}+{\bf{w}}[/latex] is the vector with an initial point that coincides with the initial point of [latex]{\bf{v}}[/latex], and with a terminal point that coincides with the terminal point of [latex]{\bf{w}}[/latex]
- zero vector
- the vector with both initial point and terminal point [latex](0,0)[/latex]
