Vector & Scalar Product Solver

In three-dimensional Cartesian coordinates, we use a set of three mutually perpendicular unit vectors called the unit triad:

• $\mathbf{i}$ is the unit vector in the positive x-direction: $\mathbf{i} = (1, 0, 0)$.
• $\mathbf{j}$ is the unit vector in the positive y-direction: $\mathbf{j} = (0, 1, 0)$.
• $\mathbf{k}$ is the unit vector in the positive z-direction: $\mathbf{k} = (0, 0, 1)$.

Any vector $\mathbf{V}$ in 3D space can be expressed as a linear combination of these unit vectors:

$$ \mathbf{V} = V_x\mathbf{i} + V_y\mathbf{j} + V_z\mathbf{k} $$

Where $V_x, V_y,$ and $V_z$ are the scalar components of $\mathbf{V}$ along the x, y, and z axes, respectively. This is equivalent to the component form $\mathbf{V} = (V_x, V_y, V_z)$.

Properties:

• $|\mathbf{i}| = |\mathbf{j}| = |\mathbf{k}| = 1$
• They are mutually orthogonal (perpendicular): $\mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} = 0$
• Dot products with themselves: $\mathbf{i} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{j} = \mathbf{k} \cdot \mathbf{k} = 1$

The scalar product (or dot product) of two vectors $\mathbf{A}$ and $\mathbf{B}$ is a scalar quantity defined as:

$$ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta $$

where $\theta$ is the angle between the two vectors ($0 \le \theta \le \pi$).

In component form, if $\mathbf{A} = A_x\mathbf{i} + A_y\mathbf{j} + A_z\mathbf{k}$ and $\mathbf{B} = B_x\mathbf{i} + B_y\mathbf{j} + B_z\mathbf{k}$, the dot product is calculated as:

$$ \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z $$

Properties:

• Commutative: $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$
• Distributive: $\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$
• $\mathbf{A} \cdot \mathbf{A} = |\mathbf{A}|^2$
• If $\mathbf{A} \cdot \mathbf{B} = 0$ (and $\mathbf{A}, \mathbf{B}$ are non-zero), then $\mathbf{A}$ and $\mathbf{B}$ are orthogonal (perpendicular).

The direction cosines of a vector $\mathbf{V} = V_x\mathbf{i} + V_y\mathbf{j} + V_z\mathbf{k}$ are the cosines of the angles between the vector and the positive coordinate axes.

Let $\alpha, \beta,$ and $\gamma$ be the angles that $\mathbf{V}$ makes with the positive x, y, and z axes, respectively. Then the direction cosines are:

• $l = \cos \alpha = \frac{V_x}{|\mathbf{V}|}$
• $m = \cos \beta = \frac{V_y}{|\mathbf{V}|}$
• $n = \cos \gamma = \frac{V_z}{|\mathbf{V}|}$

Where $|\mathbf{V}| = \sqrt{V_x^2 + V_y^2 + V_z^2}$ is the magnitude of the vector.

An important property is that the sum of the squares of the direction cosines is always equal to 1:

$$ l^2 + m^2 + n^2 = \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 $$

A unit vector in the direction of $\mathbf{V}$ can be written using the direction cosines: $\mathbf{\hat{v}} = l\mathbf{i} + m\mathbf{j} + n\mathbf{k}$.

The scalar product has several important applications in physics and engineering:

• Work Done: If a constant force $\mathbf{F}$ moves an object through a displacement $\mathbf{d}$, the work done $W$ by the force is $W = \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}| |\mathbf{d}| \cos \theta$.
• Projection of a Vector: The scalar projection of vector $\mathbf{A}$ onto vector $\mathbf{B}$ is given by $\text{comp}_{\mathbf{B}}\mathbf{A} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{B}|} = |\mathbf{A}| \cos \theta$. The vector projection is $(\text{comp}_{\mathbf{B}}\mathbf{A}) \frac{\mathbf{B}}{|\mathbf{B}|}$.
• Power: The instantaneous power $P$ delivered by a force $\mathbf{F}$ acting on an object moving with velocity $\mathbf{v}$ is $P = \mathbf{F} \cdot \mathbf{v}$.
• Checking Orthogonality: As mentioned, if $\mathbf{A} \cdot \mathbf{B} = 0$, the vectors are perpendicular.

The vector product (or cross product) of two vectors $\mathbf{A}$ and $\mathbf{B}$ is a vector quantity, denoted $\mathbf{A} \times \mathbf{B}$, defined as follows:

• Magnitude: $|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin \theta$, where $\theta$ is the angle between $\mathbf{A}$ and $\mathbf{B}$ ($0 \le \theta \le \pi$).
• Direction: The direction of $\mathbf{A} \times \mathbf{B}$ is perpendicular to the plane containing $\mathbf{A}$ and $\mathbf{B}$, determined by the right-hand rule (point fingers of right hand along $\mathbf{A}$, curl towards $\mathbf{B}$, thumb points in direction of $\mathbf{A} \times \mathbf{B}$).

In component form, using the determinant notation:

$$ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_y B_z - A_z B_y)\mathbf{i} - (A_x B_z - A_z B_x)\mathbf{j} + (A_x B_y - A_y B_x)\mathbf{k} $$

Properties:

• Anti-commutative: $\mathbf{A} \times \mathbf{B} = -(\mathbf{B} \times \mathbf{A})$
• Distributive: $\mathbf{A} \times (\mathbf{B} + \mathbf{C}) = (\mathbf{A} \times \mathbf{B}) + (\mathbf{A} \times \mathbf{C})$
• $\mathbf{A} \times \mathbf{A} = \mathbf{0}$ (the zero vector)
• If $\mathbf{A} \times \mathbf{B} = \mathbf{0}$ (and $\mathbf{A}, \mathbf{B}$ are non-zero), then $\mathbf{A}$ and $\mathbf{B}$ are parallel or anti-parallel ($\theta = 0$ or $\theta = \pi$).
• $\mathbf{i} \times \mathbf{j} = \mathbf{k}$, $\mathbf{j} \times \mathbf{k} = \mathbf{i}$, $\mathbf{k} \times \mathbf{i} = \mathbf{j}$
• $\mathbf{j} \times \mathbf{i} = -\mathbf{k}$, $\mathbf{k} \times \mathbf{j} = -\mathbf{i}$, $\mathbf{i} \times \mathbf{k} = -\mathbf{j}$

The vector product is crucial in many areas of physics and geometry:

• Torque (Moment of a Force): The torque $\boldsymbol{\tau}$ produced by a force $\mathbf{F}$ acting at a position $\mathbf{r}$ relative to a pivot point is $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$.
• Area of a Parallelogram: The magnitude of the cross product, $|\mathbf{A} \times \mathbf{B}|$, is equal to the area of the parallelogram formed by vectors $\mathbf{A}$ and $\mathbf{B}$ when placed tail-to-tail.
• Angular Momentum: The angular momentum $\mathbf{L}$ of a particle with linear momentum $\mathbf{p}$ at position $\mathbf{r}$ relative to an origin is $\mathbf{L} = \mathbf{r} \times \mathbf{p}$.
• Magnetic Force: The force $\mathbf{F}_B$ on a charge $q$ moving with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}$ is given by the Lorentz force equation: $\mathbf{F}_B = q(\mathbf{v} \times \mathbf{B})$.
• Finding a Normal Vector: The cross product $\mathbf{A} \times \mathbf{B}$ gives a vector that is normal (perpendicular) to the plane containing $\mathbf{A}$ and $\mathbf{B}$.

A line in 3D space can be defined by a point it passes through and a direction vector parallel to the line.

Let $\mathbf{r}_0$ be the position vector of a known point $P_0$ on the line, and let $\mathbf{v}$ be a vector parallel to the line (the direction vector).

The position vector $\mathbf{r}$ of any point $P$ on the line can be expressed as:

$$ \mathbf{r} = \mathbf{r}_0 + t\mathbf{v} $$

where $t$ is a scalar parameter. As $t$ varies over all real numbers, the point $P$ traces out the entire line.

If $\mathbf{r} = (x, y, z)$, $\mathbf{r}_0 = (x_0, y_0, z_0)$, and $\mathbf{v} = (a, b, c)$, the vector equation can be written in parametric form:

• $x = x_0 + ta$
• $y = y_0 + tb$
• $z = z_0 + tc$

If $a, b, c$ are all non-zero, we can also write the symmetric equations:

$$ \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} (= t) $$