📘 vector geometry

1. **Problem Statement:** Prove using vectors that the angle subtended by a diameter of a circle at any point on the semicircle is a right angle (90 degrees). 2. **Setup and Notati

1. **Problem statement:** We want to find points $P$ on the line $g$ defined by $\vec{r} = t \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$ such that the angle $\angle APB = 90^\circ$,

1. **State the problem:** We have a parallelogram ACBD with vectors \(\overrightarrow{OA} = a\) and \(\overrightarrow{OB} = b\).

1. **State the problem:** We are given a parallelogram ABCD with vector \(\overrightarrow{BC} = \binom{-a}{-b}\), where \(a > 0\) and \(b > 0\). The gradient (slope) of BC is 1, an

1. **State the problem:** We are given two lines:

1. **State the problem:** We have points $P(2,1,3)$, $Q(4,3,4)$, and $R(2,1,8)$. Point $N$ is the foot of the perpendicular from $P$ to the line joining $Q$ and $R$. We need to fin

1. **Problem statement:** Given quadrilateral ABCD with points O, A, B, C, D and vectors satisfying \(\overrightarrow{OA} = 2 \overrightarrow{CB}\), N is the midpoint of segment AD

1. **Stating the problem:** We are given three points in 3D space: $A(-2,1,6)$, $B(2,4,5)$, and $C(-1,-2,1)$. We need to show that these points form a right triangle and then find

1. Bài toán yêu cầu xác định đẳng thức đúng liên quan đến tích vô hướng của các vectơ \(\overrightarrow{OA}\) và \(\overrightarrow{OB}\), với \(M\) là trung điểm của đoạn thẳng \(A

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