Formulas for Vectors

Some of the most important formulas for vectors such as the magnitude, the direction, the unit vector, addition, subtraction, scalar multiplication and cross product are presented.

Vector Defined by two Points

\( \) \( \)\( \) \( \) The components of a vector \( \vec {PQ} \) defined by two points \( P(P_x \;, \; P_y \;, \; P_z )\) (initial point) and \( Q(Q_x \;, \; Q_y \;, \; Q_z )\) (terminal point) are given as follows: \[ \vec{PQ} = \;< Q_x - P_x \;, \; Q_y - P_y \;, \; Q_z - P_z > \]

In what follows \( \vec A, \vec B \) and \( \vec C \) are 3 dimensional vectors given by their components as follows:
\( \vec A = \; < A_x \;, \; A_y \;, \; A_z > \)
\( \vec B = \; < B_x \;, \; B_y \;, \; B_z > \)
\( \vec C = \; < C_x \;, \; C_y \;, \; C_z > \)

Magnitude of a Vector

The magnitude of vector \( \vec A \) written as \( |\vec A| \) is given by
\[ |\vec A| = \sqrt{A_x^2 + A_y^2 + A_z^2} \]

Unit Vector

A unit vector is a vector whose magnitude is equal to 1.
The unit vector \( \vec u \) that has the same direction as vector \( \vec A \) is given by
\[ \vec u = \dfrac{\vec A}{|\vec A|} = \; < \dfrac{A_x}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \;, \; \dfrac{A_y}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \;,\; \dfrac{A_z}{\sqrt{A_x^2 + A_y^2 + A_z^2}} > \]

Direction of a Vector

In 3 dimensional space, the direction of a vector is defined by 3 angles \( \alpha \) , \( \beta \) and \( \gamma\) (see Figure 1. below) called direction cosines.

cosine directions of a 3d vector
Figure 1. - Direction cosine of a vector.
These are the angles between the vector and the positive x-, y- and z- axes respectively of a rectangular system. The cosine of these angles, for vector \( \vec A \) ,are given by:
\[ \cos (\alpha) = \dfrac{A_x}{|\vec A|} = \dfrac{A_x}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \] \[ \cos (\beta) = \dfrac{A_y}{|\vec A|} = \dfrac{A_y}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \] \[ \cos (\gamma) = \dfrac{A_z}{|\vec A|} = \dfrac{A_z}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \]

In 2-D, the direction of a vector is defined as an angle ( angle \( \theta \) in the figure below ) that a vector makes with the positive x-axis.

angle of a 2-D vector
Figure 2. - Angle of a 2-D vector.
Vector \( \vec A = \; < A_x \;, \; A_y > \) is given by \[ \theta = \arctan (\dfrac{A_y}{A_x}) \] taking into account the signs of \( A_x \) and \( A_y \) to determine the quadrant where the terminal side of the vector is located.

Operations on Vectors

  • Addition

    The addition of vectors \( \vec A \) and \( \vec B \) is defined by \[ \vec A + \vec B = \; < A_x + B_x \; , \; A_y + B_y \; , \; A_z + B_z > \] More on Vector Addition.

  • Subtraction

    The subtraction of vectors \( \vec A \) and \( \vec B \) is defined by \[ \vec A - \vec B = \; < A_x - B_x \; , \; A_y - B_y \; , \; A_z - B_z > \] More on vector subtraction and adding and subtracting vectors.

  • Multiply Vector by a Scalar

    The multiplication of a vector \( \vec A \) by a scalar \( k \) is defined by \[ k \vec A = \; < k A_x \; , \; k A_y \; , \; k A_z > \]

Scalar Product of Vectors

Definition

The Scalar (or dot) product of two vectors \( \vec A \) and \( \vec B \) is given by \[ \vec A \cdot \vec B = |\vec A| \cdot |\vec B| \cdot\cos \theta \] where \( \theta \) is the angle between vectors \( \vec A \) and \( \vec B \).
Given the coordinates of vectors \( \vec A \) and \( \vec B \), it can be shown that \[ \vec A \cdot \vec B = A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z \]

Properties of Scalar Product

\[ \vec A \cdot \vec B = \vec B \cdot \vec A \] \[ \vec A \cdot (\vec B + \vec C) = \vec A \cdot \vec B + \vec A \cdot \vec C \] \[ k \vec A \cdot (\vec B) = k (\vec A \cdot \vec B) \]

Orthogonal Vectors

Two vectors \( \vec A \) and \( \vec B \) are orthogonal, the angle \( \theta \) between equal to \( 90^{\circ} \), if and only if \[ \vec A \cdot \vec B = |\vec A| \cdot |\vec B| \cdot \cos \theta = |\vec A| \cdot |\vec B| \cdot \cos 90^{\circ} = 0\]

Angle Between Two Vectors

If \( \theta \) is the angle made by two vectors \( \vec A \) and \( \vec B \), then \[ \cos \theta = \dfrac{\vec A \cdot \vec B}{ |\vec A|\cdot |\vec B|} = \dfrac{A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z}{ \sqrt{A_x^2 + A_y^2 + A_z^2} \cdot\sqrt{B_x^2 + B_y^2 + B_z^2}} \]

Cross Product

The cross product of two vectors \( \vec A \) and \( \vec B \) is a vector orthogonal to both vectors and is given by
\[ \vec A \times \vec B = \begin{vmatrix} \vec i & \vec j & \vec k\\ A_x & A_y & A_z\\ B_x & B_y & B_z \end{vmatrix} \\ = (A_y B_z - A_z B_y ) \vec i - (A_x B_z - A_z B_x) \vec j + (A_x B_y - A_y B_x) \vec k\]

Properties of Cross Product

\[ \vec A \times \vec B = - \vec B \times \vec A \] \[ (k \vec A) \times \vec B = \vec A \times (k \vec B ) = k( \vec A \times \vec B) \] The cross product is a vector and there may a need as in eletromagnetism and many other topics in physics to find the orientation of this vector. Use the right hand rule to find the orientation of the cross product: point the index in the direction of vector A, the middle finger in the direction of vector B and the direction of the cross product A × B is in the same direction of the thumb. right hand for cross product

Geometrical Meaning of Cross Product

The area of a parallelogram defined by vectors \( \vec A \) and \( \vec B \) is the magnitude of their cross product given by: Parallelogram Formed by Two Vectors \[ \text{Area of Parallelogram} = |\vec A \times \vec B| = |\vec A | \cdot |\vec B| \cdot |\sin \theta|\]

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