Dependence and independence. Bases
In the plane, two non-parallel vectors were enough to describe any other vector. In space you need one more: with only two, however independent they are, you can never leave the plane that they form themselves. This section answers two questions: when are three vectors really independent? And if they are, how do you use them to give coordinates to any other vector?
Dependence and independence of two vectors
Two surveillance drones move through space along straight trajectories defined by the vectors $\vec{u} = (2, -4, 6)$ and $\vec{v} = (-1, 2, -3)$. Determine whether the…
Two surveillance drones move through space along straight trajectories defined by the vectors $\vec{u} = (2, -4, 6)$ and $\vec{v} = (-1, 2, -3)$. Determine whether the drones move in parallel directions by analysing the linear dependence of their direction vectors.
A laser sensor system requires two beams of light to be parallel. The first beam follows the direction of vector $\vec{d_1} = (3, k, -1)$ and the second follows vector…
A laser sensor system requires two beams of light to be parallel. The first beam follows the direction of vector $\vec{d_1} = (3, k, -1)$ and the second follows vector $\vec{d_2} = (6, -4, -2)$. Find the value of parameter $k$ so that the vectors are linearly dependent and, therefore, the beams are parallel.
The determinant: dependence of three vectors
Given the vectors $\vec{u} = (1, 2, 3)$, $\vec{v} = (2, 0, 1)$ and $\vec{w} = (3, 2, 4)$, prove without performing complex calculations (by observing the relationship…
Given the vectors $\vec{u} = (1, 2, 3)$, $\vec{v} = (2, 0, 1)$ and $\vec{w} = (3, 2, 4)$, prove without performing complex calculations (by observing the relationship between their components) that the set $\{\vec{u}, \vec{v}, \vec{w}\}$ is linearly dependent.
Check whether the vectors $\vec{e_1} = (1, 0, 0)$, $\vec{e_2} = (1, 1, 0)$ and $\vec{e_3} = (1, 1, 1)$ can form a basis of $\mathbb{R}^3$. To do this, justify whether…
Check whether the vectors $\vec{e_1} = (1, 0, 0)$, $\vec{e_2} = (1, 1, 0)$ and $\vec{e_3} = (1, 1, 1)$ can form a basis of $\mathbb{R}^3$. To do this, justify whether they are linearly independent using the determinant of the matrix they form.
State with reasons whether the following statement is true or false: "In three-dimensional space $\mathbb{R}^3$, any set of four non-zero vectors is always linearly…
State with reasons whether the following statement is true or false: "In three-dimensional space $\mathbb{R}^3$, any set of four non-zero vectors is always linearly dependent." Give a simple example to support your answer.
A basis of space
In a video game graphics engine, the position of an object is defined by a linear combination of two basic vectors: $\vec{a} = (1, 0, 2)$ and $\vec{b} = (0, -1, 3)$.…
In a video game graphics engine, the position of an object is defined by a linear combination of two basic vectors: $\vec{a} = (1, 0, 2)$ and $\vec{b} = (0, -1, 3)$. Calculate the coordinates of the resulting vector $\vec{w}$ if the following operation is applied:
$$\vec{w} = 3\vec{a} - 2\vec{b}$$If a vector $\vec{v}$ has coordinates $(3, -2, 5)$ with respect to the standard basis $\{\vec{i}, \vec{j}, \vec{k}\}$, express it as a linear combination of these unit…
If a vector $\vec{v}$ has coordinates $(3, -2, 5)$ with respect to the standard basis $\{\vec{i}, \vec{j}, \vec{k}\}$, express it as a linear combination of these unit vectors.
Coordinates with respect to another basis
Consider the points in space $A(1, 0, 0)$, $B(0, 2, 1)$, and $C(2, 0, m)$. a) Find the vector $\vec{AB}$ and the vector $\vec{AC}$. b) Determine the value of $m$ such…
Consider the points in space $A(1, 0, 0)$, $B(0, 2, 1)$, and $C(2, 0, m)$.
- a) Find the vector $\vec{AB}$ and the vector $\vec{AC}$.
- b) Determine the value of $m$ such that the vector $\vec{v} = (3, -3, -3)$ is linearly dependent on $\vec{AB}$ and $\vec{AC}$.
- c) For the value of $m$ found, write $\vec{v}$ as a linear combination of $\vec{AB}$ and $\vec{AC}$.
Given the vectors of the standard basis $\{\vec{i}, \vec{j}, \vec{k}\}$, the following vectors are defined: $$\vec{u} = \vec{i} + \vec{j} \quad , \quad \vec{v} = \vec{j}…
Given the vectors of the standard basis $\{\vec{i}, \vec{j}, \vec{k}\}$, the following vectors are defined:
$$\vec{u} = \vec{i} + \vec{j} \quad , \quad \vec{v} = \vec{j} + \vec{k} \quad , \quad \vec{w} = \vec{i} + \vec{k}$$Justify whether the set $\{\vec{u}, \vec{v}, \vec{w}\}$ is a basis of $\mathbb{R}^3$. If a vector $\vec{x}$ has coordinates $(1, 2, 3)$ in the standard basis, what will its coordinates be in the basis $B' = \{\vec{u}, \vec{v}, \vec{w}\}$?
Back to the start: Three vectors are independent if their determinant is non-zero ($\det(\vec u,\vec v,\vec w)\neq 0$), which guarantees that they are not coplanar; from them, any vector $\vec x$ can be expressed uniquely by solving the system $\vec x=a\vec u+b\vec v+c\vec w$ to find its coordinates $(a,b,c)$ in that basis.