📘 vector geometry

1. **Énoncé du problème :** Nous devons montrer que pour le point N, intersection de la droite (CD) avec (AB), on a $$\overrightarrow{AN} = \frac{2}{3} \overrightarrow{AB}$$.

1. **State the problem:** Find the distance between point $C(1,2,0)$ and the line $p$ given by the parametric equation $$\mathbf{r}(t) = (3,0,1) + t(1,-1,3), \quad t \in \mathbb{R}

1. **Problem statement:** Given points $A(3,1,2)$, $B(0,1,-2)$, and $C(1,2,0)$, find point $D$ such that $ABCD$ forms a parallelogram. 2. **Formula and concept:** In vector geometr

1. **Problem statement:** We have triangle OAB with vectors: $\vec{OA} = 8\vec{c}$, $\vec{OB} = 4\vec{d}$, $\vec{BP} = 2\vec{d}$, and $\vec{OM} = 6\vec{c}$. Point N is the midpoint

1. **Problem statement:** We have two lines in 3D space:

1. **Problem statement:** Find the equations of planes $P_1$ and $P_2$, the angle between them, the vector equation of their line of intersection, and the distance of this line fro

1. **Problem statement:** We have quadrilateral OABC with vectors $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$. Points M and N lie on lines OB and AB respectively, with ratios $OM

1. **Problem statement:** We have a parallelogram $OABC$ with vectors $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$. Point $M$ lies on $BC$ such that $B

1. **Problem statement:** Find the image $A'$ of the point $A(2,1,2)$ in the line given by the parametric form $\mathbf{r} = \mathbf{a} + \lambda \mathbf{d}$ where $\mathbf{a} = (1

1. **Problem statement:** We have triangle OAB with points P and N on OA and OB respectively, M is midpoint of AB, and lines OPM and APN are straight. Given $OP : PM = 4 : 3$, find

1. **Stating the problem:** We will explore vector translation, vector addition and subtraction both graphically and algebraically, scalar multiplication of vectors, calculation of

1. **Problem Statement:** Find the ratio in which a point divides a line segment and determine the midpoint of the segment. 2. **Formula for Ratio:** If a point $P(x,y)$ divides th

1. **Problem statement:** Given a parallelogram ABCD and points M, N, P, Q defined by $$\overrightarrow{AM} = \frac{3}{2} \overrightarrow{AB}, \quad \overrightarrow{BN} = \frac{3}{

1. **Problem statement:** We have a parallelogram WXYZ with vectors \(\overrightarrow{WX} = \mathbf{u}\) and \(\overrightarrow{WZ} = \mathbf{v}\). M is the midpoint of \(\overright

1. **State the problem:** We have a parallelogram OXYZ with vectors \(\overrightarrow{OX} = a\) and \(\overrightarrow{OY} = b\). Points \(P\) and \(R\) divide \(OX\) and \(OY\) in

1. **State the problem:** We are given vectors \(\overrightarrow{AB} = -3\mathbf{i} + 6\mathbf{j}\) and \(\overrightarrow{AC} = 10\mathbf{i} - 2\mathbf{j}\) in triangle ABC.

1. **Problem statement:** Given vectors \(\overrightarrow{AB} = -3i + 6j\) and \(\overrightarrow{AC} = 10i - 2j\) in triangle ABC, find: (a) The size of angle \(\angle BAC\) in deg

1. **State the problem:** Given vectors \( \vec{AB} = -3\mathbf{i} + 6\mathbf{j} \) and \( \vec{AC} = 10\mathbf{i} - 2\mathbf{j} \), find: (a) The size of angle \( \angle BAC \) in

1. **Stating the problem:** We have triangle OAB with vectors \(\mathbf{a} = \overrightarrow{OA}\) and \(\mathbf{b} = \overrightarrow{OB}\).

1. Given points and vectors in a coordinate system: $O(0,0)$, $I(1,0)$, $J(0,1)$ with $OI=OJ=1$ cm. 2. Points provided: $A(-3,0)$, $M(3,2)$, and $N(3,-2)$.

1. **Problem Statement:** You are given a roof with rectangular base OABC where OA = 14 m along unit vector i, OC = 8 m along unit vector j, and the top edge DE is 6 m long and 5 m