Straight Lines

Passes Through a Point & Parallel to a Vector

Equation for a line that:

• passes through $\underline{r_0}=\langle x_0, y_0, z_0 \rangle$
• is parallel to $\underline{v}=a\underline{i}+b\underline{j}+c\underline{k}$

Parametric Equation

$\underline{r}=\underline{r_0}+t\underline{v};\;t\in\mathbb{R}$

Symmetric Equation

Passes Through 2 Points

Equation of a line passes through $A=(x_1,y_1,z_1)$, $B=(x_2,y_2,z_2)$. $\underline{r_A}$ and $\underline{r_B}$ are the position vectors of $A$ and $B$.

$\underline{r}=(1-t)\underline{r_A}+t\underline{r_B};\;t\in\mathbb{R}$

Intersection

To show that two straight lines intersect in 3D space, existence of a point which satisfies both lines must be proven.

It is not enough to show that the cross product of their parallel vectors is non-zero.

Normal to 2 Lines

Let $\alpha,\beta$ be two lines.

Here $v_1=\langle a_1, b_1, c_1\rangle$, $v_2=\langle a_2, b_2, c_2\rangle$ are $2$ vectors parallel to $\alpha, \beta$ respectively.

Normal to both lines: $v_1 \times v_2$.

Unit Normal

Angle Between 2 Straight Lines

Using the $\alpha,\beta$ lines mentioned above:

Here $v_1, v_2$ are $2$ vectors parallel to $\alpha, \beta$ respectively.

Shortest Distance from a Point

The distance can be calculated using Pythogoras’ theorem.