Revision: Line and Plane Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

If a plane at a distance p from the origin and a unit normal \[\hat{n}\]

\[\overline{r}\cdot\hat{n}-p=0\]

Cartesian normal form:

lx + my + nz = p

Normals: n₁, n₂

\[\cos\theta=\left|\frac{\overline{n}_1\cdot\overline{n}_2}{\left|\overline{n}_1\right|\cdot\left|\overline{n}_2\right|}\right|\]

If line direction vector = b
Plane normal = n

\[\sin\theta=\left|\frac{\overline{b}\cdot\overline{n}}{\left|\overline{b}\right|\cdot\left|\overline{n}\right|}\right|\]

Lines:

\[\overline{r}=\overline{a}_{1}+\lambda_{1}\overline{b}_{1}\]

\[\overline{r}=\overline{a}_2+\lambda_2\overline{b}_2\]

Condition for Coplanarity:

\[\left(\overline{a}_{2}-\overline{a}_{1}\right)\cdot\left(\overline{b}_{1}\times\overline{b}_{2}\right)=0\]

Cartesian Condition:

\[\begin{vmatrix}
x-x_1 & y-y_1 & z-z_1 \\
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2
\end{vmatrix}=0\]

If line: r = a + λb 

Distance of point P(α) from the line:

\[\sqrt{\left|\overline{\alpha}-\overline{a}\right|^2-\left[\frac{\left(\overline{\alpha}-\overline{a}\right)\cdot\overline{b}}{\left|\overline{b}\right|}\right]^2}\]

\[d=\frac{|(a_2-a_1)\cdot(b_1\times b_2)|}{|b_1\times b_2|}\]

\[|(\overline{a_2}-\overline{a_1})\times\hat{b}|\]

Plane:

\[\bar{r}\cdot\hat{n}=p\]

The distance of the origin from the plane = \[\left|p-{\bar{a}}\cdot{\hat{n}}\right|\]

1.Line through point \[\mathrm{A}(\overline{a})\] and parallel to vector b

 \[\overline{r}=\overline{a}+\lambda\overline{b}\]

2. line passing through two points \[\mathrm{A}(\bar{a})\] and \[\mathrm{B}(\bar{b})\]

\[\overline{r}=\overline{a}+\lambda(\overline{b}-\overline{a})\]

1.Through point \[\mathrm{A}(\overline{a})\] and normal vector n

\[\overline{r}\cdot\overline{n}=\overline{a}\cdot\overline{n}\]

2. Cartesian Form

\[a\left(x-x_{1}\right)+b(y-y_{1})+c(z-z_{1})=0\]

3. Plane Through Three Non-Collinear Points

Vector Form:

\[\left(\overline{r}-\overline{a}\right)\cdot\left(\overline{b}-\overline{a}\right)\times\left(\overline{c}-\overline{a}\right)=0\]

Cartesian form:

\[\begin{vmatrix}
x-x_1 & y-y_1 & z-z_1 \\
x_2-x_1 & y_2-y_1 & z_2-z_1 \\
x_3-x_1 & y_3-y_1 & z_3-z_1
\end{vmatrix}=0\]

1.Through A(x₁, y₁, z₁) with direction ratios a, b, c

\[\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\]

• \[x=x_1+\lambda a\]
• \[y=y_1+\lambda b\]
• \[z=z_1+\lambda c\]

2. Through two points A(x₁,y₁,z₁), B(x₂,y₂,z₂)

\[\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}\]