Key Points

• 1. Watch the video:

• 2. Watch this solved problem video

Key Points

• 1. Watch the video:

• 2. Watch this solved problem video

Key Points

• 1. Watch the video:

• 2. Perpendicular vectors and dot product. Watch this video

• 3. Example of perpendicular vectors. Watch this video

Key Points

• 1. Watch the video:

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

Key Points

• 1. Watch the video:

• 2. Perpendicular vectors to a line and distance. Watch this video

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