Further Maths - Vectors
See also:
- [[Further Maths - Vector Equation of a Line]]A
- [[Further Maths - Vector Equation of a Plane]]A
- [[Further Maths - Dot Product]]A
What is the formula for the distance to a three dimensional point $(a,b,c)$?
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\[\left(\begin{matrix} a \\ b \\ c \end{matrix}\right)\]What is the formula for the length of the vector??
\[\sqrt{a^2 + b^2 + c^2}\]#####
\[\pmb{a} = \left(\begin{matrix} a _ 1 \\ a _ 2 \\ a _ 3 \end{matrix}\right) \\ \pmb{b} = \left(\begin{matrix} b _ 1 \\ b _ 2 \\ b _ 3 \end{matrix}\right)\]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?
\[\pmb{a} = \left(\begin{matrix} a _ 1 \\ a _ 2 \\ a _ 3 \end{matrix}\right) \\ \pmb{b} = \left(\begin{matrix} b _ 1 \\ b _ 2 \\ b _ 3 \end{matrix}\right)\]
What is the formula for the distance between the two vectors?
What is the vector parallel to the $x$-axis?
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??
\[\left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right)\]#####
\[\left(\begin{matrix} x \\ y \\ z \end{matrix}\right) \to \left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right)\]When does this trick not work?? When the value of $x$ is actually $0$.
What is the vector parallel to the $y$-axis?
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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?
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?
\[\left(\begin{matrix} x \\ y \\ z \end{matrix}\right) \to \left(\begin{matrix} 1 \\\\ y \\\\ z \end{matrix}\right)\]
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?
\[x + y + z = 0\\5x - 2y + 3z = 4\]
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 $1$?
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 $\pmb{\hat{X}}$?
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?
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