📘 vector algebra

1. Let's state the problem: You want to find the intersection point of two vectors (or lines) without using a parameter like $\lambda$. 2. Typically, vector intersection problems i

1. **Problem statement:** We have trapezium OACB with vectors \(\overrightarrow{OA} = 2\mathbf{a}\), \(\overrightarrow{AB} = 5\mathbf{b}\), and \(\overrightarrow{AC} = 3\mathbf{b}\

1. **Problem statement:** Given a triangle ABC with vectors \(\vec{a} = \overrightarrow{AB}\) and \(\vec{b} = \overrightarrow{AC}\), find the vectors represented by: (i) \(\overrig

1. **Problem Statement:** Find the area of the parallelogram whose adjacent sides are given by vectors \(\vec{a} = \hat{i} - \hat{j} + 3 \hat{k}\) and \(\vec{b} = 2\hat{i} - 7\hat{

1. لنبدأ بكتابة المعادلة المعطاة: $$\overrightarrow{BC'} = 3 \overrightarrow{BA'}$$

1. **Problem statement:** Given triangle OAB with points P on AB and C on OB such that $AP : PB = 2 : 3$ and $OC : CB = 1 : 2$, find the values of $r$ and $s$ in the vector express

1. **Problem statement:** Given points on triangle $ABC$ with $D$ on $AB$ such that $\overrightarrow{AD} : \overrightarrow{DB} = 2 : 1$ and $E$ on $AC$ such that $\overrightarrow{A

1. **State the problem:** Find the shortest distance between the two lines given by their vector equations: $$\vec{r_1} = (1 - t)\hat{i} + (t - 2)\hat{j} + (3 - 2t)\hat{k}$$

1. **State the problem:** Find the magnitude and direction of the vector $\begin{pmatrix}5 \\ 2\end{pmatrix}$. 2. **Magnitude formula:** The magnitude $|\mathbf{v}|$ of a vector $\

1. **State the problem:** We are given two vector equations of lines in 3D:

1. **State the problem:** Convert the Cartesian equation of the line $$\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2}$$ into its vector form. 2. **Recall the formula:** The ve

1. **State the problem:** Show that the three lines with direction cosines

1. **Problem Statement:** We want to find the vector joining two points $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$. 2. **Formula Used:** The vector from $P_1$ to $P_2$ is given by

1. **Problem Statement:** Find the sum of all values of $\beta$ for which the points represented by position vectors $$\vec{A} = 2\hat{i} + 3\hat{j} + \hat{k}, \quad \vec{B} = 2\ha

1. Let's start by stating the problem: We want to understand the rules for the dot product and cross product of vectors. 2. **Dot product rule:** The dot product of two vectors $\m

1. **Problem statement:** Find the component of vector $\vec{B}$ perpendicular to vector $\vec{A}$ given that $|\vec{A}| = 3$. 2. **Recall:** The component of $\vec{B}$ perpendicul

1. **Problem statement:** Given points and vectors with relationships:

1. **Problem I (a): Construct points E, F, and L in parallelogram ABCD** Given ABCD is a parallelogram, and points E and F are defined by vectors:

1. **Problem 1: Find $6 \overline{A} + \overline{B}$** Given vectors:

1. The problem is to find the magnitude of the vector $\mathbf{v} = 5\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}$.\n\n2. The formula for the magnitude of a vector $\mathbf{v} = a\mathbf

1. The problem is to simplify the given vector expressions for \( \vec{PQ} \) and express them in simplest form. 2. We will analyze each option and simplify the coefficients where