📘 vector algebra

1. The problem asks to find the vector $\vec{AB}$ given points $A(-2,6)$ and $B(8,-5)$.\n\n2. The formula for vector $\vec{AB}$ is $\vec{AB} = (x_B - x_A)\hat{i} + (y_B - y_A)\hat{

1. **Problem statement:** Given two vectors in 3D space $\vec{u} = (2, -1, 3)$ and $\vec{v} = (1, 2, -2)$, find the coordinates of the vector $\vec{w} = [\vec{u}, \vec{v}]$, which

1. **Problem:** Given two vectors $\vec{a} = (1, 2, -1)$ and $\vec{b} = (3, 0, 2)$ in space $Oxyz$, find the coordinates of the cross product vector $[\vec{a}, \vec{b}]$. 2. **Form

1. **Stating the problem:** We are given two displacement vectors for a drone's journey: \(\overrightarrow{AB} = (4,8,6)\) km and \(\overrightarrow{BC} = (-1,-4,6)\) km. We need to

1. **State the problem:** We need to find the area of a parallelogram formed by two vectors \( \mathbf{a} = [-5,1,3] \) and \( \mathbf{b} = [-2,3,-4] \). 2. **Formula:** The area o

1. **Problem 1(a):** Given vectors \( \overrightarrow{PQ} = \begin{pmatrix} -3 \\ 7 \end{pmatrix} \) and \( 4\overrightarrow{PR} = \begin{pmatrix} -2 \\ 8 \end{pmatrix} \), find \(

1. **Problem statement:** Given quadrilateral OABC with vectors \(\vec{OA} = \vec{a}\), \(\vec{AB} = 3\vec{b} + \frac{1}{2}\vec{a}\), and \(\vec{OC} = 2\vec{b}\). Point D lies on O

1. **Stating the problem:** We are given quadrilateral OABC with vectors:

1. **State the problem:** We are given two vectors \( \vec{OA} = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k} \) and \( \vec{OB} = 4\mathbf{i} + 8\mathbf{j} + 5\mathbf{k} \). We want to

1. **Problem statement:** Determine if the point $P(1|3|0)$ lies on the line passing through points $A(3|2|0)$ and $B(-1|4|0)$.

1. **Énoncé du problème :** Calculer $||\overrightarrow{AB}||$ et en déduire $||\overrightarrow{OC}||$ pour les points $A(4, -3, 1)$, $B(-2, 0, -1)$ et $C(3, -4, 5)$.

1. The problem asks for the angle between two parallel vectors. 2. Recall that parallel vectors point in the same or exactly opposite directions.

1. **State the problem:** We are given points $A(-1,2)$, $B(5,-2)$, $C(1,3)$, and $D(0,0)$. We want to express the vector $\overrightarrow{BA}$ as a linear combination of vectors $

1. Planteamos el problema: En un hexágono regular, se nos da el vector \(\overrightarrow{CX} = -3\mathbf{u} + 2\mathbf{v} + \frac{3}{2} \mathbf{w}\) y queremos hallar el punto \(X\

1. **Stating the problem:** We want to understand why the condition for a vector $w$ to be equidistant from vectors $u$ and $v$ is given by

1. **State the problem:** We are given four vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ with the sum $\mathbf{a} + \mathbf{b} + \mathbf{c} + \mathbf{d} = \la

1. **State the problem:** Convert the vector $\mathbf{A} = 3\mathbf{a}_x + 4\mathbf{a}_y$ into cylindrical components at $\theta = 53.1^\circ$ and find the radial component $A_r$.

1. **Problem statement:** In parallelogram ABCD, point M is the midpoint of side BC. Given vectors $\overrightarrow{AB} = \vec{a}$ and $\overrightarrow{AD} = \vec{b}$, find vectors

1. **Problem statement:** In parallelogram ABCD, point M is the midpoint of side BC. Given vectors \(\overline{AB} = \vec{a}\) and \(\overline{AD} = \vec{b}\), express vectors \(\o

1. **State the problem:** Find the direction angle $\theta$ of the vector $\vec{v} = (-3, -10)$ measured from the positive x-axis. 2. **Formula:** The direction angle $\theta$ is g

1. **State the problem:** Given vectors $a$ and $b$ with magnitudes $|a|=15$, $|b|=20$, and $|a-b|=15.5$, find the magnitude $|a+b|$. 2. **Recall the formula for the magnitude of t