# Numbers | Maths with Lemon

Source: https://mathswithlemon.io/aa-geometry-trigonometry-vector%20equation%20of%20a%20line.html


## Vector equation of a line

"Many people who have not studied mathematics confuse it with arithmetic and consider it a dry and fruitless science. In reality, however, it is a science which requires a great amount of imagination."


#### Vector equations of a line in 2D


##### What you have to know:

- Addition / subtraction of vectors, position vector, magnitude and direction of a vector (If not, check this lesson)
- You must be able to rearrange formulae. (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Watch this solved problem video

#### Vector equation of a line in 3D


##### What you have to know:

- Vector equation of a line in 2D.

##### Key Points

- 1. Watch the video:
- 2. Watch this solved problem video

#### Scalar product


##### What you have to know:

- How to calculate the magnitude of a vector.
- How to find an angle using the inverse function of cosine.

##### Key Points

- 1. Watch the video:
- 2. Perpendicular vectors and dot product. Watch this video
- 3. Example of perpendicular vectors. Watch this video

#### Cross product of two vectors


##### What you have to know:

- How to calculate the scalar product of two vectors.
- How to find an angle using the inverse function of cosine.

##### Key Points

- 1. Watch the video:
- 2. How to find the area of a triangle using the vector product. Watch this video

#### Angle between two lines


##### What you have to know:

- How to calculate the scalar product of two vectors.
- How to find an angle using the inverse function of cosine.

##### Key Points

- 1. Watch the video:
- 2. Perpendicular vectors to a line and distance. Watch this video

#### Extra


##### Material and references:

- Hodder Book HL(ISBN: 9781510462366) : 8D, 8E, 8F, 8G, 8H

##### Key Points

- 1. You should know that the scalar (dot) product is linked to the angle between two vectors: \[ \mathbf{a}\cdot\mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3. \]
- 2. You should be able to use the algebraic properties of the scalar product: \( \mathbf{a}\cdot\mathbf{b} = \mathbf{b}\cdot\mathbf{a} \), \( (-\mathbf{a})\cdot\mathbf{b} = -(\mathbf{a}\cdot\mathbf{b}) \), >\( \mathbf{a}\cdot(\mathbf{b}+\mathbf{c}) = (\mathbf{a}\cdot\mathbf{b})+(\mathbf{a}\cdot\mathbf{c}) \), \( (k\mathbf{a})\cdot\mathbf{b} = k(\mathbf{a}\cdot\mathbf{b}) \), and \( \mathbf{a}\cdot\mathbf{a} = |\mathbf{a}|^2 \).
- 3. You should know that vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel if \( \mathbf{b} = t\mathbf{a} \) for some scalar \( t \), and perpendicular if \( \mathbf{a}\cdot\mathbf{b} = 0 \).
- 4. You should be able to find and use various forms of the equation of a line. \[ \mathbf{r} = \mathbf{a} + \lambda\mathbf{d}, \] where \( \mathbf{d} \) is a direction vector and \( \mathbf{a} \) is the position vector of a point on the line.
- 5. Writing \( \mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \) gives the parametric equations of the line by expressing \( x, y, z \) in terms of \( \lambda \). Eliminating \( \lambda \) gives the Cartesian form \[ \frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}. \] 6. Line (motion): \( \mathbf{r}=\mathbf{a}+\lambda\mathbf{d} \), with \( \lambda=t \), velocity \( \mathbf{d} \), speed \( |\mathbf{d}| \). Angle between lines: \( \cos\theta=\dfrac{\mathbf{d}_1\cdot\mathbf{d}_2}{|\mathbf{d}_1||\mathbf{d}_2|} \). Cross product: \( \mathbf{a}\times\mathbf{b}= \begin{pmatrix} a_2b_3-a_3b_2\\ a_3b_1-a_1b_3\\ a_1b_2-a_2b_1 \end{pmatrix} \), \( |\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin\theta \) (area of parallelogram). 7. Properties of cross product: \( \mathbf{a}\times\mathbf{b} = -\,\mathbf{b}\times\mathbf{a} \) \( (k\mathbf{a})\times\mathbf{b} = \mathbf{a}\times(k\mathbf{b}) = k(\mathbf{a}\times\mathbf{b}) \) \( \mathbf{a}\times(\mathbf{b}+\mathbf{c}) = \mathbf{a}\times\mathbf{b} + \mathbf{a}\times\mathbf{c} \) parallel \( \Rightarrow \mathbf{a}\times\mathbf{b}=0 \) perpendicular \( \Rightarrow |\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}| \) \( \mathbf{r}=\mathbf{a}+\lambda\mathbf{d}_1+\mu\mathbf{d}_2 \) \( \mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n} \) \( n_1x+n_2y+n_3z=d \) line–plane angle \( = 90^\circ-\phi \) plane–plane angle \( = \) angle between normals intersection: substitute line into plane planes: parallel or intersect in a line three planes \( \Rightarrow \) point, line, or none
- 6. Line (motion): \( \mathbf{r}=\mathbf{a}+\lambda\mathbf{d} \), with \( \lambda=t \), velocity \( \mathbf{d} \), speed \( |\mathbf{d}| \). Angle between lines: \( \cos\theta=\dfrac{\mathbf{d}_1\cdot\mathbf{d}_2}{|\mathbf{d}_1||\mathbf{d}_2|} \). Cross product: \( \mathbf{a}\times\mathbf{b}= \begin{pmatrix} a_2b_3-a_3b_2\\ a_3b_1-a_1b_3\\ a_1b_2-a_2b_1 \end{pmatrix} \), \( |\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin\theta \) (area of parallelogram).
- 7. Properties of cross product: \( \mathbf{a}\times\mathbf{b} = -\,\mathbf{b}\times\mathbf{a} \) \( (k\mathbf{a})\times\mathbf{b} = \mathbf{a}\times(k\mathbf{b}) = k(\mathbf{a}\times\mathbf{b}) \) \( \mathbf{a}\times(\mathbf{b}+\mathbf{c}) = \mathbf{a}\times\mathbf{b} + \mathbf{a}\times\mathbf{c} \) parallel \( \Rightarrow \mathbf{a}\times\mathbf{b}=0 \) perpendicular \( \Rightarrow |\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}| \) \( \mathbf{r}=\mathbf{a}+\lambda\mathbf{d}_1+\mu\mathbf{d}_2 \) \( \mathbf{r}\cdot\mathbf{n}=\mathbf{a}\cdot\mathbf{n} \) \( n_1x+n_2y+n_3z=d \) line–plane angle \( = 90^\circ-\phi \) plane–plane angle \( = \) angle between normals intersection: substitute line into plane planes: parallel or intersect in a line three planes \( \Rightarrow \) point, line, or none

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