# Further Maths - Vectors

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/vectors/ · Updated: 2021-01-11 · Tags: further-maths, vectors, latex-block-alt

See also:
- [Further Maths - Vector Equation of a Line](https://ollybritton.com/notes/a-level/further-maths/topics/vector-equation-of-a-line/)
- [Further Maths - Vector Equation of a Plane](https://ollybritton.com/notes/a-level/further-maths/topics/vector-equation-of-a-plane/)
- [Further Maths - Dot Product](https://ollybritton.com/notes/a-level/further-maths/topics/vector-dot-product/)

### Magnitude and distance
##### What is the formula for the distance to a three dimensional point $(a,b,c)$??
$$
\sqrt{a^2 + b^2 + c^2}
$$
^distance-to-3d-point

##### $$ \left(\begin{matrix} a \\ b \\ c \end{matrix}\right) $$ What is the formula for the length of the vector??
$$
\sqrt{a^2 + b^2 + c^2}
$$
^length-of-vector

##### $$\pmb{a} = \left(\begin{matrix} a_1 \\ a_2 \\ a_3 \end{matrix}\right) \\  \pmb{b} = \left(\begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix}\right) $$ What is the formula for the distance between the two vectors??
$$
\sqrt{(b_1 - a_1)^2 + (b_2 - a_2)^2 + (b_3 - a_3)^2}
$$
^distance-between-two-vectors

### Axis-parallel vectors
##### What is the vector parallel to the $x$-axis??
$$
\left(\begin{matrix} 1 \\\\ 0 \\\\ 0 \end{matrix}\right)
$$
^vector-parallel-to-x-axis

##### What is the vector parallel to the $y$-axis??
$$
\left(\begin{matrix} 0 \\\\ 1 \\\\ 0 \end{matrix}\right)
$$
^vector-parallel-to-y-axis

##### What is the vector parallel to the $z$-axis??
$$
\left(\begin{matrix} 0 \\\\ 0 \\\\ 1 \end{matrix}\right)
$$
^vector-parallel-to-z-axis

### Scaling direction vectors
##### $$ \left(\begin{matrix} x \\ y \\ z \end{matrix}\right) $$ If this is a direction vector, how could you eliminate one of the unknowns??
$$
\left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right)
$$
^eliminate-unknown-in-direction-vector

##### $$ \left(\begin{matrix} x \\ y \\ z \end{matrix}\right) \to \left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right) $$ When does this trick not work??
When the value of $x$ is actually $0$.
^scaling-trick-fails-when-x-zero

### Finding a perpendicular vector
##### If you have two vectors $\pmb{a}$ and $\pmb{b}$ and you wish to find a vector $\pmb{c}$ that is perpendicular to both, what must be true??
$$
\pmb{a} \cdot \pmb{c} = 0 \\\\
\pmb{b} \cdot \pmb{c} = 0
$$
^perpendicular-vector-dot-conditions

##### $$x + y + z = 0\\5x - 2y + 3z = 4$$ Why can't you solve these two equations??
Because there are 3 unknowns but only 2 equations.
^underdetermined-two-equations-three-unknowns

##### $$ \left(\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}\right) $$ How could you write this vector for $x$ equal to $1$??
$$
\left(\begin{matrix} 1 \\\\ \frac{1}{2} \\\\ \frac{3}{2} \end{matrix}\right)
$$
^scale-vector-to-unit-x

##### What is the technique for finding a vector perpendicular vector to two other vectors??
Use the fact the dot product must be equal to zero to find and solve two simultaneous equations.
^perpendicular-vector-technique

### Unit vectors
##### What does $\pmb{\hat{X}}$ mean??
Unit/normalised vector; in the same direction as $\pmb{X}$ but has magnitude $1$.
^hat-notation-meaning

##### What is the formula for $\pmb{\hat{X}}$??
$$
\pmb{\hat{X}} = \frac{X}{|X|}
$$
^unit-vector-formula

### The equation of a sheaf
##### Suppose you have $$Ax + By + Cz = D \\ \alpha x + \beta y + \gamma z = \delta$$ These planes intersect at a sheaf. What's the general formula for a new plane that also passes through this sheaf??
$$
(Ax + By + Cz - D) + t(\alpha x + \beta y + \gamma z - \delta) = 0
$$
^sheaf-general-plane-formula

##### Suppose you have $$Ax + By + Cz = D \\ \alpha x + \beta y + \gamma z = \delta$$ These planes intersect at a sheaf. What is the first step in finding the equation of the sheaf??
Making the substitution $z = \lambda$.
^sheaf-first-step-substitution

##### Suppose you have $$Ax + By + Cz = D \\ \alpha x + \beta y + \gamma z = \delta$$ and you have made the substitution $z = \lambda$ to get $$Ax + By = D - C\lambda \\ \alpha x + \beta y = \delta - \gamma \lambda$$ What is the next step??
Solving these equations in general to come up with

$$

$$
^sheaf-next-step-solve-generally

##### Explain the general process for finding the equation of a sheaf??
$$
\left(\begin{matrix} x \\\\ y \\\\ z \end{matrix}\right) = ...
$$
^sheaf-general-process

##### If you've made the substitution $z = \lambda$ in order to find the equation of a sheaf, what will always be the value of $z$ in the final vector for the parametric form of the line??
$$
\lambda
$$
^sheaf-final-z-is-lambda

### Cautions with plane formulas
##### Why must you be careful using the $$\frac{|a\alpha + b\beta + c\gamma - d|}{\sqrt{a^2 + b^2 + c^2}}$$ formula??
Because you subtract $d$ which can be confusing if $d$ is negative.
^distance-formula-negative-d-caution

##### Why must you be careful using the "let $z = \lambda$" approach to finding the line of intersection between two planes??
Because it's very easy to write the final vector in the wrong order.
^line-of-intersection-vector-order-caution

### Reflections in a plane
##### What's the thought process for finding the reflection of a point in a plane??
Find a line that passes through the point and has the normal vector of the plane, and double the parameter for the point of intersection with the plane.
^reflection-of-point-in-plane

##### What's the thought process for finding the reflection of a line in a plane??
Create a new line out of the point of intersection with the plane and the reflection of one point in the line.
^reflection-of-line-in-plane

### Angles and exam presentation
##### How could you show two lines lie in the same plane??
Show they intersect.
^show-two-lines-coplanar

##### How can you ensure you get acute angles when using the scalar product to find angles between vectors??
Take the modulus of what you're putting into $\arccos$.
^acute-angles-via-modulus

##### Do you need to do any subtracting to work out the angle between two planes after you've worked out the angle between their normals??
No.
^angle-between-planes-no-subtraction

##### $$\cos \theta = \left| \frac{\pmb{a}\cdot\pmb{b}}{|\pmb{a}||\pmb{b}|} \right|$$ Why are the modulus signs around this important??
It ensures you only get acute angles.
^modulus-signs-ensure-acute-angle

##### $$8x + 4y - 2z = 20$$ How do they always want you to write your answers for direction vectors and planes in an exam??
As simple as possible, divide through by 2
^simplify-plane-equation-in-exam

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