Cards In This Deck

• $\text{How do you prove that three points }A,B,\text{ and }C\text{ are collinear (lie on the same straight line)?}$
  $$\begin{gathered} \text{Show that }\overrightarrow{AB}=k\overrightarrow{BC}\text{ (where }k\text{ is a scalar).} \\ \text{AND state that they share a common point (}B\text{).} \end{gathered}$$
• $\text{Condition to equate coefficients of vectors on both sides of an equation (e.g., if }p\mathbf{a}+q\mathbf{b}=r\mathbf{a}+s\mathbf{b}\text{, then }p=r\text{ and }q=s\text{)?}$
  $\text{The vectors }\mathbf{a}\text{ and }\mathbf{b}\text{ must be non-parallel.}$
• $\text{How do you find the unit vector in the direction of vector }\mathbf{a}\text{?}$
  $\hat{\mathbf{a}}=\frac{\mathbf{a}}{|\mathbf{a}|}$
• $$\begin{gathered} \text{True or False?} \\ \text{To find the vector }\overrightarrow{AB}\text{, you calculate }\mathbf{a}-\mathbf{b}\text{.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }\overrightarrow{AB}=\mathbf{b}-\mathbf{a} \\ \text{(Position vector of End Point minus Position vector of Start Point)} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{When calculating the scalar product }\mathbf{a}\cdot \mathbf{b}\text{ using components, you include the }\mathbf{i},\mathbf{j},\mathbf{k}\text{ unit vectors in the final answer.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: The result is a Scalar (a number), not a vector.} \end{gathered}$$
• $\text{When finding the angle }\angle ABC\text{ using scalar products, which vectors must you use?}$
  $$\begin{gathered} \text{Use }\overrightarrow{BA}\text{ and }\overrightarrow{BC}\text{.} \\ \text{(Vectors must be ’Tail-to-Tail’ or ’Head-to-Head’ at the vertex where the angle is).} \end{gathered}$$
• $\text{How do you find the area of a triangle }ABC\text{ using vectors?}$
  $$\begin{gathered} \text{1. Calculate magnitudes }|\overrightarrow{AB}|\text{ and }|\overrightarrow{AC}|\text{.} \\ \text{2. Use scalar product }\overrightarrow{AB}\cdot \overrightarrow{AC}\text{ to find angle }A\text{.} \\ \text{3. Use Area }=\frac{1}{2}|\overrightarrow{AB}||\overrightarrow{AC}|\sin A\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If the scalar product }\mathbf{a}\cdot \mathbf{b}=0\text{, then the vectors are parallel.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: They are perpendicular (orthogonal).} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{In the line equation }\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}\text{, the vector }\mathbf{a}\text{ must be the point where the line crosses the y-axis.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }\mathbf{a}\text{ can be the position vector of any fixed point on the line.} \end{gathered}$$
• $\text{Formula for the vector equation of a line passing through points }A\text{ and }B\text{?}$
  $$\begin{gathered} \mathbf{r}=\mathbf{a}+\lambda (\mathbf{b}-\mathbf{a}) \\ \text{Where }\mathbf{a}\text{ is the position vector of }A\text{ and }\mathbf{b}\text{ is the position vector of }B\text{.} \end{gathered}$$
• $\text{You have determined that two lines in 3D are not parallel and do not intersect. What term describes them?}$
  $\text{Skew lines.}$
• $$\begin{gathered} \text{Procedure to determine if two lines }{\mathbf{r}}_1={\mathbf{a}}_1+\lambda {\mathbf{b}}_1\text{ and }{\mathbf{r}}_2={\mathbf{a}}_2+ \\ \mu {\mathbf{b}}_2\text{ intersect?} \end{gathered}$$
  $$\begin{gathered} \text{1. Equate components (}x,y,z\text{) to form 3 simultaneous equations.} \\ \text{2. Solve two equations to find }\lambda \text{ and } \\ \mu \text{.} \\ \text{3. Check if these values satisfy the third equation.} \end{gathered}$$