📘 vector algebra

1. **State the problem:** Determine which pairs of lines are skew lines. Skew lines are lines that are not parallel and do not intersect. 2. **Recall definitions and formulas:**

1. **State the problem:** We have two vectors \( \vec{G} \) and \( \vec{H} \) in 3D space with magnitudes and angles given. We need to find:

1. **Problem statement:** We have a regular hexagon centered at the origin O.

1. **Problem statement:** Given that $\overrightarrow{AB} = 2a + 18b$ and point $P$ lies on line $AB$ such that the ratio $AP : PB = 1 : 3$, find the vector $\overrightarrow{AP}$ i

1. **State the problem:** Given vector $\overrightarrow{AB} = -7\mathbf{i} + 3\mathbf{j} + 8\mathbf{k}$ and point $A(-2, -3, 6)$, find the coordinates of point $B$. 2. **Recall the

1. **State the problem:** Find a vector of magnitude 7 in the direction of \( \mathbf{v} = 12\mathbf{i} - 5\mathbf{k} \). 2. **Formula and rules:** To find a vector of a given magn

1. **Problem statement:** Given vectors $a$, $b$, and $c$ such that $a \neq 0$ and $a \times 3b = 2a \times c$, show that $3b - 2c = \lambda a$ for some scalar $\lambda$. 2. **Reca

1. **Stating the problem:** We are given a rectangular prism (balok) with vertices labeled as follows:

1. We check each exercise for correctness by verifying the vector calculations and parameter substitutions. 2. For the first exercise (points on line AB):

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.