📘 vector algebra

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

1. The problem is to calculate the angle between the vectors $\vec{a} = (1, 2)$ and $\vec{c} = (-2, 5)$. 2. The formula to find the angle $\theta$ between two vectors $\vec{u}$ and

1. **State the problem:** We are given vectors \(\overline{A} = (4, -6)\), \(\overline{B} = (1, 4)\), and \(\overline{P} = (-1, -15)\). We want to verify or find \(\overline{P}\) u

1. **State the problem:** We have two unit vectors $\mathbf{A}$ and $\mathbf{B}$ such that their sum is $\mathbf{A} + \mathbf{B} = 1$. We want to find the dot product $\mathbf{A} \

1. **State the problem:** We need to find the sum of two vectors $\mathbf{s}$ and $\mathbf{t}$, where $\mathbf{s} = \langle 1, -4 \rangle$ and $\mathbf{t} = \langle -2, 5 \rangle$.

1. **State the problem:** Calculate the dot product $a \cdot b$ for the given vectors. 2. **Recall the dot product formula:** For vectors $a = \langle a_1, a_2 \rangle$ and $b = \l

1. **Problem statement:** Find two unit vectors orthogonal to the given vectors $\mathbf{a}$ and $\mathbf{b}$. 2. **Formula and concept:** Two vectors are orthogonal if their dot p

1. **State the problem:** We need to compute the cross product $\mathbf{a} \times \mathbf{b}$ where $\mathbf{a} = \langle 3, 0, -1 \rangle$ and $\mathbf{b} = \langle 1, 2, 2 \rangl

1. **State the problem:** Find the vector sum $\vec{AB} + \vec{BC} + \vec{CA}$ where $A$, $B$, and $C$ are vertices of a triangle.