Further Maths - Vectors


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What is the formula for the distance to a three dimensional point $(a,b,c)$?


\[\sqrt{a^2 + b^2 + c^2}\]

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\[\left(\begin{matrix} a \\ b \\ c \end{matrix}\right)\]

What is the formula for the length of the vector??

\[\sqrt{a^2 + b^2 + c^2}\]

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\[\pmb{a} = \left(\begin{matrix} a _ 1 \\ a _ 2 \\ a _ 3 \end{matrix}\right) \\ \pmb{b} = \left(\begin{matrix} b _ 1 \\ b _ 2 \\ b _ 3 \end{matrix}\right)\]

What is the formula for the distance between the two vectors??

\[\sqrt{(b_1 - a_1)^2 + (b_2 - a_2)^2 + (b_3 - a_3)^2}\]
\[\left(\begin{matrix} a \\ b \\ c \end{matrix}\right)\]

What is the formula for the length of the vector?


\[\sqrt{a^2 + b^2 + c^2}\]
\[\pmb{a} = \left(\begin{matrix} a _ 1 \\ a _ 2 \\ a _ 3 \end{matrix}\right) \\ \pmb{b} = \left(\begin{matrix} b _ 1 \\ b _ 2 \\ b _ 3 \end{matrix}\right)\]

What is the formula for the distance between the two vectors?


\[\sqrt{(b _ 1 - a _ 1)^2 + (b _ 2 - a _ 2)^2 + (b _ 3 - a _ 3)^2}\]

What is the vector parallel to the $x$-axis?


\[\left(\begin{matrix} 1 \\\\ 0 \\\\ 0 \end{matrix}\right)\]

2021-01-20

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\[\left(\begin{matrix} x \\ y \\ z \end{matrix}\right)\]

If this is a direction vector, how could you eliminate one of the unknowns??

\[\left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right)\]

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\[\left(\begin{matrix} x \\ y \\ z \end{matrix}\right) \to \left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right)\]

When does this trick not work?? When the value of $x$ is actually $0$.

What is the vector parallel to the $y$-axis?


\[\left(\begin{matrix} 0 \\\\ 1 \\\\ 0 \end{matrix}\right)\]

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\[x + y + z = 0\\5x - 2y + 3z = 4\]

Why can’t you solve these two equations?? Because there are 3 unknowns but only 2 equations.

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\[\left(\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}\right)\]

How could you write this vector for $x$ equal to $1$??

\[\left(\begin{matrix} 1 \\\\ \frac{1}{2} \\\\ \frac{3}{2} \end{matrix}\right)\]

What is the vector parallel to the $z$-axis?


\[\left(\begin{matrix} 0 \\\\ 0 \\\\ 1 \end{matrix}\right)\]

2021-05-17

\[\left(\begin{matrix} x \\ y \\ z \end{matrix}\right)\]

If this is a direction vector, how could you eliminate one of the unknowns?


\[\left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right)\]
\[\left(\begin{matrix} x \\ y \\ z \end{matrix}\right) \to \left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right)\]

When does this trick not work?


When the value of $x$ is actually $0$.

2021-09-20

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??

\[(Ax + By + Cz - D) + t(\alpha x + \beta y + \gamma z - \delta) = 0\]
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 $z = \lambda$.

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

\[\]

If you have two vectors $\pmb{a}$ and $\pmb{b}$ and you wish to find a vector $\pmb{c}$ that is perpindicular to both, what must be true?


\[\pmb{a} \cdot \pmb{c} = 0 \\\\ \pmb{b} \cdot \pmb{c} = 0\]
\[x + y + z = 0\\5x - 2y + 3z = 4\]

Why can’t you solve these two equations?


Because there are 3 unknowns but only 2 equations.

2022-01-19

Why must you be careful using the
\[\frac{ \vert a\alpha + b\beta + c\gamma - d \vert }{\sqrt{a^2 + b^2 + c^2}}\]

formula?? Because you subtract $d$ which can be confusing if $d$ is negative.

\[\left(\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}\right)\]

How could you write this vector for $x$ equal to $1$?


\[\left(\begin{matrix} 1 \\\\ \frac{1}{2} \\\\ \frac{3}{2} \end{matrix}\right)\]

What is the technique for finding a vector perpindicular vector to two other vectors?


Use the fact the dot product must be equal to zero to find and solve two simulatenous equations.

What does $\pmb{\hat{X}}$ mean?


Unit/normalised vector; in the same direction as $\pmb{X}$ but has magnitude $1$.

2022-05-17

What is the formula for $\pmb{\hat{X}}$?


\[\pmb{\hat{X}} = \frac{X}{ \vert X \vert }\]

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?


\[(Ax + By + Cz - D) + t(\alpha x + \beta y + \gamma z - \delta) = 0\]

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 $z = \lambda$.

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\[\cos \theta = \left \vert \frac{\pmb{a}\cdot\pmb{b}}{ \vert \pmb{a} \vert \vert \pmb{b} \vert } \right \vert\]

Why are the modulus signs around this important?? It ensures you only get acute angles.

2022-05-17

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\[8x + 4y - 2z = 20\]

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




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