Vectors and vector spaces
In 1st Bachillerato you worked with vectors in the plane: two numbers were enough to describe them. But a drone does not only move forwards and sideways, it also goes up and down; a mechanical part does not only have length and width, it also has height. To describe real space you need a third coordinate, and almost everything you already know about vectors extends naturally.
Vectors with three coordinates
Express the vector $\vec{w} = (7, 11, 2)$ as a linear combination of the standard basis vectors of $\mathbb{R}^3$, that is, in terms of $\vec{i} = (1, 0, 0)$,…
Express the vector $\vec{w} = (7, 11, 2)$ as a linear combination of the standard basis vectors of $\mathbb{R}^3$, that is, in terms of $\vec{i} = (1, 0, 0)$, $\vec{j} = (0, 1, 0)$ and $\vec{k} = (0, 0, 1)$.
The vector $\vec{AB}$ and its magnitude
A surveillance drone takes off from a platform located at point $A(1, -2, 5)$ and travels in a straight line to reach a target at point $B(4, 3, -1)$. Calculate the…
A surveillance drone takes off from a platform located at point $A(1, -2, 5)$ and travels in a straight line to reach a target at point $B(4, 3, -1)$. Calculate the components of the displacement vector $\vec{AB}$ and obtain its magnitude to find the distance travelled by the drone.
Addition and multiplication by a scalar
Given the vectors $\vec{u} = (2, -1, 3)$ and $\vec{v} = (0, 4, 1)$, perform the following analytical operations: a) $\vec{w} = \vec{u} + \vec{v}$ b)…
Given the vectors $\vec{u} = (2, -1, 3)$ and $\vec{v} = (0, 4, 1)$, perform the following analytical operations:
- a) $\vec{w} = \vec{u} + \vec{v}$
- b) $\vec{t} = 3\vec{u} - 2\vec{v}$
- c) Calculate the value of the scalar $k$ if it is known that $k \cdot \vec{u} = (10, -5, 15)$.
Linear combinations and dependence
Determine whether the vectors $\vec{a} = (1, -2, 4)$ and $\vec{b} = (-3, 6, -12)$ are linearly dependent. Justify your answer by checking if their coordinates are…
Determine whether the vectors $\vec{a} = (1, -2, 4)$ and $\vec{b} = (-3, 6, -12)$ are linearly dependent. Justify your answer by checking if their coordinates are proportional and explain what geometric relationship exists between them (are they parallel?).
Calculate the value of the parameter $m$ so that the vector $\vec{u} = (m, 2, 4)$ is a linear combination of the vectors $\vec{v} = (1, 0, 2)$ and $\vec{w} = (0, 1, -1)$…
Calculate the value of the parameter $m$ so that the vector $\vec{u} = (m, 2, 4)$ is a linear combination of the vectors $\vec{v} = (1, 0, 2)$ and $\vec{w} = (0, 1, -1)$ through the expression $\vec{u} = 3\vec{v} + 2\vec{w}$.
Points, position vectors and applications
In the design of a metallic structure, two beams are joined at their ends represented by the points $P(2, 5, -3)$ and $Q(8, -1, 7)$. To reinforce the structure, a…
In the design of a metallic structure, two beams are joined at their ends represented by the points $P(2, 5, -3)$ and $Q(8, -1, 7)$. To reinforce the structure, a support must be placed exactly at the midpoint of line segment $PQ$. Find the coordinates of this midpoint $M$.
Three radio stations are located at points $A(1, 1, 1)$, $B(3, 2, 0)$ and $C(5, 3, -1)$. Check whether the three stations are collinear (belong to the same straight…
Three radio stations are located at points $A(1, 1, 1)$, $B(3, 2, 0)$ and $C(5, 3, -1)$. Check whether the three stations are collinear (belong to the same straight line) by studying the proportionality of the vectors $\vec{AB}$ and $\vec{BC}$.
Consider a parallelogram where three of its consecutive vertices are $A(1, 1, 0)$, $B(4, 2, 1)$ and $C(5, 5, 3)$. Using the equality between vectors (for example,…
Consider a parallelogram where three of its consecutive vertices are $A(1, 1, 0)$, $B(4, 2, 1)$ and $C(5, 5, 3)$. Using the equality between vectors (for example, $\vec{AB} = \vec{DC}$), calculate the coordinates of the fourth vertex $D$.
Equations of a line in space
A line is determined by a point $A(a_1,a_2,a_3)$ and a non-zero direction vector $\vec v=(v_1,v_2,v_3)$. Its vector equation is $\vec{OP}=\vec{OA}+t\vec v$, $t\in\mathbb R$. In components:
$$x=a_1+tv_1,\qquad y=a_2+tv_2,\qquad z=a_3+tv_3.$$These are the parametric equations. If all three components of the direction vector are non-zero, solving for t gives the continuous (symmetric) form:
$$\frac{x-a_1}{v_1}=\frac{y-a_2}{v_2}=\frac{z-a_3}{v_3}.$$If a component is zero, keep that coordinate constant: do not divide by zero. For the line through two distinct points A and B you can take $\vec v=\vec{AB}$.
The parameter runs through positions along a fixed direction:
Choose the form according to the task: parametric to generate points or model motion; two planes to solve intersections using systems. To extract the direction vector from two intersecting planes, the cross product of their normals works.