Formulae [7]
Vector:
Angle between two lines: \[\cos\theta=\left|\frac{\mathbf{b}_{1}\cdot\mathbf{b}_{2}}{|\mathbf{b}_{1}||\mathbf{b}_{2}|}\right|\]
If two lines are perpendicular: b₁ · b₂ = 0
If two lines are parallel: b₁ = λb₂
Cartesian:
\[\cos\theta=\frac{|a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}|}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\]
If two lines are perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0
If two lines are parallel: \[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\]
Vector Form:
The length of the perpendicular from a point P (α) to the line \[\overline{\mathrm{r}}=\overline{\mathrm{a}}+\lambda\overline{\mathrm{b}}\] is
\[\sqrt{\left|\overline{\alpha}-\overline{a}\right|^2-\left[\frac{(\overline{\alpha}-\overline{a}).\overline{b}}{\left|\overline{b}\right|}\right]^2}\]
Cartesian Form:
The length of the perpendicular from the point P (a, b, c) on the line \[\frac{x-x_{1}}{l}=\frac{y-y_{1}}{\mathrm{m}}=\frac{\mathrm{z-z_{1}}}{\mathrm{n}}\] is \[\sqrt{[(\mathrm{a-x_{1}})^{2}+(\mathrm{b-y_{1}})^{2}+(\mathrm{c-z_{1}})^{2}]-[(\mathrm{a-x_{1}})l+(\mathrm{b-y_{1}})\mathrm{m+(c-z_{1})n}]^{2}}\] where l, m, n are direction cosines of line.
\[SD=\left|\frac{\left(a_{2}-a_{1}\right)\times b}{\left|b\right|}\right|\]
Vector Form:
\[\mathbf{d}=\left|\frac{(\overline{\mathbf{b}}_{1}\times\overline{\mathbf{b}}_{2}).(\overline{\mathbf{a}}_{2}-\overline{\mathbf{a}}_{1})}{\left|\overline{\mathbf{b}}_{1}\times\overline{\mathbf{b}}_{2}\right|}\right|\]
Cartesian Form:
\[\mathbf{d}=\left|\frac{ \begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ \mathbf{a}_1 & \mathbf{b}_1 & \mathbf{c}_1 \\ \mathbf{a}_2 & \mathbf{b}_2 & \mathbf{c}_2 \end{vmatrix}}{\sqrt{\left(\mathbf{a}_1\mathbf{b}_2-\mathbf{a}_2\mathbf{b}_1\right)^2+\left(\mathbf{a}_1\mathbf{c}_2-\mathbf{a}_2\mathbf{c}_1\right)^2+\left(\mathbf{b}_1\mathbf{c}_2-\mathbf{b}_2\mathbf{c}_1\right)^2}}\right|\]
Vector Form:
\[\cos\theta=\left|\frac{\overline{\mathbf{n₁}}.\overline{\mathbf{n₂}}}{\left|\overline{\mathbf{n₁}}\right|.\left|\overline{\mathbf{n₂}}\right|}\right|\]
Cartesian Form:
\[\cos\theta=\left|\frac{\mathrm{a}_{1}\mathrm{a}_{2}+\mathrm{b}_{1}\mathrm{b}_{2}+\mathrm{c}_{1}\mathrm{c}_{2}}{\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}}\sqrt{\mathrm{a}_{2}^{2}+\mathrm{b}_{2}^{2}+\mathrm{c}_{2}^{2}}}\right|\]
Vector Form:
\[\sin\theta=\left|\frac{\overline{\mathbf{b}}.\overline{\mathbf{n}}}{\left|\overline{\mathbf{b}}\right|.\left|\overline{\mathbf{n}}\right|}\right|\]
Cartesian Form:
\[\mathrm{sin}\theta=\frac{\mathrm{aa}_{1}+\mathrm{bb}_{1}+\mathrm{cc}_{1}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}}\sqrt{\mathrm{a}_{1}^{2}+\mathrm{b}_{1}^{2}+\mathrm{c}_{1}^{2}}}\]
Vector Form:
\[\mathbf{d}=\frac{\left|\left(\overline{\mathbf{a}}.\overline{\mathbf{n}}\right)-\mathbf{p}\right|}{\left|\overline{\mathbf{n}}\right|}\]
Cartesian Form:
\[\mathbf{d}=\left|\frac{\mathbf{a}x_{1}+\mathbf{b}y_{1}+\mathbf{c}z_{1}+\mathbf{d}}{\sqrt{\mathbf{a}^{2}+\mathbf{b}^{2}+\mathbf{c}^{2}}}\right|\]
Key Points
| Case | Vector Form | Cartesian Form (Symmetric Form) |
|---|---|---|
| 1. Through a point + parallel to vector | r = a + λb | x = x₁ + lλ y = y₁ + mλ z = z₁ + nλ |
| 2. Through two points | r = a + λ(b − a) | x − x₁ / (x₂ − x₁) = y − y₁ / (y₂ − y₁) = z − z₁ / (z₂ − z₁) |
| Case | Vector Form | Cartesian Form |
|---|---|---|
| 1. Normal form (given normal vector) | \[\overline{\mathbf{r}}.\hat{\mathbf{n}}=\mathbf{p}\] | ax + by + cz + d = 0 |
| 2. Through a point (x₁, y₁, z₁) | \[\begin{bmatrix} \mathbf{\overline{r}}-\mathbf{\overline{a}} \end{bmatrix}.\mathbf{\overline{n}}=\mathbf{0}\] | a(x−x₁) + b(y−y₁) + c(z−z₁) = 0 |
| 3. Through point + parallel to two vectors | \[\begin{bmatrix} \overline{\mathbf{r}}\overline{\mathbf{b}}\overline{\mathbf{c}} \end{bmatrix}= \begin{bmatrix} \overline{\mathbf{a}}\overline{\mathbf{b}}\overline{\mathbf{c}} \end{bmatrix}\] | \[\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ \mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 \\ \mathbf{c}_1 & \mathbf{c}_2 & \mathbf{c}_3 \end{vmatrix}=0\] |
| 4. Through three non-collinear points | \[(\mathbf{r-a})\cdot[(\mathbf{b-a})\times(\mathbf{c-a})]=0\] | \[\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix}=0\] |
| 5. Through the intersection of two planes | \[\left(\overline{\mathbf{r}}.\overline{\mathbf{n}}_1-\mathbf{d}_1\right)+\lambda\left(\overline{\mathbf{r}}.\overline{\mathbf{n}}_2-\mathbf{d}_2\right)=0\] | (a₁x + b₁y + c₁z + d₁) + λ(a₂x + b₂y + c₂z + d₂) = 0 |
Equation of a Plane in Intercept form:
\[\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\]
Distance of the Plane from Origin is
\[d=\frac{1}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}}\]
Condition for Perpendicularity:
\[\overline{\mathbf{b}}=\lambda\overline{\mathbf{n}}\], λ is a parameter
\[\frac{\mathbf{a}_{1}}{\mathbf{a}}=\frac{\mathbf{b}_{1}}{\mathbf{b}}=\frac{\mathbf{c}_{1}}{\mathbf{c}}\]
Condition for Parallelism:
The line is parallel to the plane, if
\[\overline{\mathbf{b}}.\overline{\mathbf{n}}=0\]
aa₁ + bb₁ + cc₁ = 0
Vector Form:
Condition for coplanarity of two lines:
Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar if
(a₁ − a₂) · (b₁ × b₂) = 0
Equation of the plane containing both lines:
\[\left(\overline{\mathbf{r}}-\overline{\mathbf{a}_1}\right).\left(\overline{\mathbf{b}_1}\times\overline{\mathbf{b}_2}\right)=\mathbf{0}\] or \[\left(\overline{\mathbf{r}}-\overline{\mathbf{a}_2}\right).\left(\overline{\mathbf{b}_1}\times\overline{\mathbf{b}_2}\right)=\mathbf{0}\]
Cartesian Form:
\[\begin{vmatrix} x_2-x_1 & y_2-y_1 & z_2-z_1 \\ \mathbf{a}_1 & \mathbf{b}_1 & \mathbf{c}_1 \\ \mathbf{a}_2 & \mathbf{b}_2 & \mathbf{c}_2 \end{vmatrix}=0\]
