📘 vector algebra

1. **State the problem:** Given vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ with $\mathbf{a} \neq \mathbf{0}$, and the equation $\mathbf{a} \times 3\mathbf{b} = 2 \mathbf{

1. **Problem statement:** Given that the cross product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is zero, i.e., $\mathbf{u} \times \mathbf{v} = \mathbf{0}$, what can we deduce a

1. **Problem:** Use the parallelogram method to add vectors $\vec{x} = \langle 3, 4 \rangle$ and $\vec{y} = \langle 4, 1 \rangle$. 2. **Formula:** Vector addition by the parallelog

1. Let's start by stating the problem: Understanding how vector calculations work and elaborating on the basic operations. 2. Vectors are quantities with both magnitude and directi

1. **State the problem:** Given points $A(-2,0,10)$, $B(1,9,3)$, and $C(2,-4,6)$, find vectors $\overrightarrow{AC}$ and $\overrightarrow{BC}$, then find the angle $\angle ACB$. 2.

1. **Problem:** Calculate the dot product (also called scalar product) of the vectors given in part 1a: $\mathbf{u} = (2, -3)$ and $\mathbf{v} = (1, 2)$. 2. **Formula:** The dot pr

1. **State the problem:** Find a unit vector perpendicular to both vectors $\mathbf{a} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{b} = -2\mathbf{i} + 3\mathbf{j} + 2\mat

1. **State the problem:** Find a unit vector perpendicular to both vectors $\mathbf{a} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{b} = -2\mathbf{i} + 3\mathbf{j} + 2\mat

1. **State the problem:** Find the angle between the vectors $\mathbf{u} = 3\mathbf{i} + 2\mathbf{j}$ and $\mathbf{v} = 5\mathbf{i} - \mathbf{j}$.\n\n2. **Formula used:** The angle

1. **State the problem:** Determine if points $A(5,7,4)$ and $B(9,12,4)$ lie on the line $L$ given by the vector equation: $$\vec{r} = \begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix} + \la

1. **State the problem:** We are given points A, B, C, and D on segment AC such that $AD : DC = 2 : 3$. We know $\overrightarrow{AB} = 10a$ and $\overrightarrow{DB} = 2a - 4b$. We

1. **State the problem:** We need to find the sum of the vectors $\langle 1, -2 \rangle$ and $\langle 1, 8 \rangle$, then find the magnitude and direction of the resultant vector.

1. **Problem Statement:** We have a satellite trajectory given by the vector equation $$\vec{r}(t) = \langle 1, 2, 0 \rangle + t \langle 2, 1, 2 \rangle$$ and a ground station at p

1. **State the problem:** We are given two vectors \( \mathbf{u} = 4 \langle \cos 55^\circ, \sin 55^\circ \rangle \) and \( \mathbf{v} = 2 \langle \cos 350^\circ, \sin 350^\circ \r

1. **Stel het probleem vast:** We moeten vectoren schrijven als veelvouden van de vector $\overrightarrow{AB}$.

1. **State the problem:** We are given a line defined by the parametric equation $\mathbf{r}(t) = (1,1,1) + t(2,1,-1)$ and three vectors $\mathbf{w} = (0,-1,1)$, $\mathbf{v} = (-2,

1. **State the problem:** We are given two points A(1, 11) and B(5, 4).

1. **Problem:** Calculate the dot product $\vec{u} \cdot \vec{v}$ and classify the angle between $\vec{u}$ and $\vec{v}$ for the vectors: (a) $\vec{u} = (2,8)$ and $\vec{v} = (-3,1

1. **Problem 1:** Given triangle OAB with vectors \(\vec{OA} = 5\vec{a}\) and \(\vec{OB} = 2\vec{b}\), point T lies on AB such that \(AT : TB = 5 : 1\). Show that \(\vec{OT}\) is p

1. **Problem statement:** In pentagon OABCD, given that OA is parallel to DC, AB is parallel to OD, with OD = 2AB and OA = 2DC, and vectors \(\overrightarrow{OA} = \mathbf{a}\) and

1. **Problem statement:** Calculate the angle between vectors $\mathbf{a} = (1,2)$ and $\mathbf{c} = (-2,5)$ in degrees. 2. **Formula:** The angle $\theta$ between two vectors $\ma