Key Concepts

1. A quantity defined by both a magnitude (such as a distance) and a direction is called a vector.
2. Two vectors are equal if they have the same length and direction; it does not matter where the vector starts.
3. The length of a vector [latex]\bf{v}[/latex] is called its magnitude, and is denoted by [latex]\|{\bf{v}}\|\text{.}[/latex]
4. The sum of two vectors [latex]\bf{u}[/latex] and [latex]\bf{v}[/latex] is a new vector, [latex]\bf{w}\text{,}[/latex] starting at the tail of the first vector and ending at the head of the second vector. The sum is called the resultant vector.
5. Addition of vectors is commutative. The rule for adding vectors is sometimes called the parallelogram rule.

6. Operations on Vectors.

   1. We can multiply a vector, [latex]\bf{v}\text{,}[/latex] by a scalar, [latex]k\text{.}[/latex]
      1. If [latex]k \gt 0\text{,}[/latex] the magnitude of [latex]k\bf{v}[/latex] is [latex]k[/latex] times the magnitude of [latex]\bf{v}\text{.}[/latex] The direction of [latex]k\bf{v}[/latex] is the same as the direction of [latex]\bf{v}\text{.}[/latex]
      2. If [latex]k \lt 0\text{,}[/latex] the direction of [latex]k\bf{v}[/latex] is opposite the direction of [latex]\bf{v}\text{.}[/latex]
   2. We can add two vectors [latex]\bf{v}[/latex] and[latex]\bf{w}[/latex] with the parallelogram rule.
7. Any vector can be written as the sum of its horizontal and vertical vector components, [latex]\bf{v_x}[/latex] and [latex]\bf{v_y}\text{.}[/latex]
8. The components of a vector [latex]\bf{v}[/latex] whose direction is given by the angle [latex]\theta[/latex] in standard position are the scalar quantities[latex]v_x = \|{\bf{v}}\| \cos \theta[/latex][latex]v_y = \|{\bf{v}}\| \sin \theta[/latex]
9. The magnitude and direction of a vector with components and are given by [latex]\|{\bf{v}}\| = \sqrt{(v_x)^2 + (v_y)^2}~~~~\text{and}~~~~\tan \theta = \dfrac{v_y}{v_x}[/latex]
10. To add two vectors using components, we can resolve each vector into its horizontal and vertical components, add the corresponding components, then compute the magnitude and direction of the resultant.
11. A vector of magnitude 1 is called a unit vector. The unit vector in the direction of the [latex]x[/latex]-axis is denoted by [latex]\bf{i}\text{.}[/latex] The unit vector in the direction of the [latex]y[/latex]-axis is called [latex]\bf{j}\text{.}[/latex]

12. Coordinate Form of a Vector.

    The vector

    [latex]{\bf{v}} = a{\bf{i}} + b{\bf{j}}[/latex]

    is the vector whose horizontal component is [latex]a[/latex] and whose vertical component is [latex]b\text{.}[/latex]

13. Comparing the Geometric and Coordinate Forms of a Vector.

    Suppose that the vector [latex]{\bf{v}}[/latex] has magnitude [latex]\|{\bf{v}}\|[/latex] and points in the direction of the angle [latex]\theta[/latex] in standard position. If [latex]{\bf{v}}[/latex] has the coordinate form [latex]{\bf{v}} = a{\bf{i}} + b{\bf{j}}\text{,}[/latex] then

    [latex]a = \|{\bf{v}}\| \cos \theta \|{\bf{v}}\| = \sqrt{a^2 + b^2}\\ b = \|{\bf{v}}\| \sin \theta \tan \theta = \dfrac{b}{a}[/latex]

14. Scalar Multiplication in Coordinate Form.

    If [latex]{\bf{v}} = a{\bf{i}} + b{\bf{j}}[/latex] and [latex]k[/latex] is a scalar, then

    [latex]k{\bf{v}} = ka{\bf{i}} + kb{\bf{j}}[/latex]

15. Sum of Vectors in Coordinate Form.

    If [latex]{\bf{u}} = a{\bf{i}}+b{\bf{j}}[/latex] and [latex]{\bf{v}} = c{\bf{i}}+d{\bf{j}}\text{,}[/latex] then

    [latex]{\bf{u}} + {\bf{v}} = (a + c){\bf{i}}+ (b + d){\bf{j}}[/latex]

16. Scaling a Vector.

    A unit vector [latex]{\bf{u}}[/latex] in the direction of [latex]{\bf{v}}[/latex] is given by [latex]{\bf{u}} = \dfrac{1}{\|{\bf{v}}\|} {\bf{v}}.[/latex]

    A vector [latex]{\bf{w}}[/latex] of length [latex]k[/latex] in the direction of [latex]{\bf{v}}[/latex] is given by [latex]{\bf{w}} = \dfrac{k}{\|{\bf{v}}\|} {\bf{v}}.[/latex]

17. Dot Product (Coordinate Formula).

    The dot product of two vectors [latex]{\bf{v}} = v_1{\bf{i}} + v_2 {\bf{j}}[/latex] and [latex]{\bf{w}} = w_1{\bf{i}} + w_2 {\bf{j}}[/latex] is the scalar

    [latex]{\bf{v}} \cdot {\bf{w}} = v_1w_1+v_2w_2[/latex]

18. The dot product is a way of multiplying two vectors that depends on the angle between them.

19. Dot Product (Geometric Formula).

    The dot product of two vectors [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex] is the scalar

    [latex]{\bf{v}} \cdot {\bf{w}} = \|{\bf{v}}\|\|{\bf{w}}\| \cos \theta[/latex]

    where [latex]\theta[/latex] is the angle between the vectors.

20. The component of a vector [latex]{\bf{w}}[/latex] in the direction of vector [latex]{\bf{v}}[/latex] is the length of the vector projection of [latex]{\bf{w}}[/latex] onto [latex]{\bf{w}}\text{.}[/latex]

21. Component of a Vector.

    The component of [latex]{\bf{w}}[/latex] in the direction of [latex]{\bf{v}}[/latex] is the scalar

    [latex]\text{comp}_{\bf{v}}{\bf{w}} = \dfrac{{\bf{v}} \cdot {\bf{w}}}{\|{\bf{v}}\|}[/latex]

22. Angle between Two Vectors.

    The angle [latex]\theta[/latex] between two vectors [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex] is given by

    [latex]\cos \theta = \dfrac{{\bf{v}} \cdot {\bf{w}}}{\|{\bf{v}}\| \|{\bf{w}}\|}[/latex]

23. Two vectors [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex] are orthogonal if [latex]{\bf{v}} \cdot {\bf{w}} = 0[/latex]