📘 vector algebra

1. **Problem statement:** Given vectors $\mathbf{A} = 2\mathbf{i} - 3\mathbf{j}$, $\mathbf{B} = -\mathbf{j} + \mathbf{k}$, and $\mathbf{C} = \mathbf{i} + \mathbf{j} + \mathbf{k}$,

1. **Problem Statement:** ABCD is a parallelogram and EF is the midpoint of side BC. Given vectors \( \mathbf{B} = (b_1, b_2) \) and \( \mathbf{F} = (c_1, c_2) \), express \( \math

1. **Problem:** Represent the vector $\mathbf{v}$ using the column approach and the $\mathbf{i}, \mathbf{j}, \mathbf{k}$ unit vector approach. 2. **Formula:** A vector $\mathbf{v}

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. **State the problem:** We are given two vectors $\mathbf{a} = 2\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ and $\mathbf{b} = 3\mathbf{i} + 4\mathbf{k}$. We need to find the direction

1. **State the problem:** We have triangle OAD with vectors \(\vec{OA} = \mathbf{a}\) and \(\vec{OD} = \mathbf{a} + \mathbf{d}\). Point C lies on line AD such that \(AC : AD = 2 :

1. The problem states: Given vectors $\vec{a} = (3, 2)$ and $\vec{b} = (-1, 0)$, find the coordinates and length of the vector $\vec{c} = \vec{a} - \vec{b}$. 2. The formula for vec

1. **Problem:** Find the cross product $\mathbf{a} \times \mathbf{b}$ where $\mathbf{a} = 2\mathbf{i} + \mathbf{j}$ and $\mathbf{b} = 3\mathbf{i} - \mathbf{k}$. 2. **Formula:** The

1. **Problem Statement:** Given points B(3,5) and C(6,-9), we need to find the vector BC, its components, and its magnitude. 2. **Vector BC:** The vector from point B to point C is

1. **Find the area of triangle ABC with vertices A(18, -2, -12), B(14, 4, -10), and C(7, 6, -6).** 2. The area of a triangle with vertices in 3D can be found using the formula:

1. The problem asks to find the vector expression for $AD$ given that $AB = \vec{a}$ and $CD = 2AB = 2\vec{a}$. We need to express $AD$ in terms of $\vec{a}$ and $\vec{b}$. 2. Reca

1. Find the components of the vector $\overrightarrow{P_1P_2}$. (a) Given $P_1 = (3, 5)$ and $P_2 = (2, 8)$, the components are found by subtracting coordinates:

1. **Problem (c):** Given vectors $\vec{QP} = 5\mathbf{i} - 3\mathbf{j}$ and $\vec{OQ} = 3\mathbf{i} + 5\mathbf{j}$, find the magnitude of $\vec{PQ}$. 2. **Recall:** The vector $\v

1. **State the problem:** We need to find the direction of the resultant vector $\vec{R}$ of three vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ with respect to the x-axis, denoted a

1. **State the problem:** Find the angle between vectors $\mathbf{u} = [-1, 3, 4]$ and $\mathbf{v} = [2, 1, -1]$. 2. **Formula:** The angle $\theta$ between two vectors $\mathbf{u}

1. The problem is to find the value of the vector expression $2\mathbf{i} \times 2\mathbf{j} \cdot \mathbf{k}$. 2. Recall the vector operations involved:

1. **State the problem:** We are given three vectors $\mathbf{u} = \mathbf{i} - 2\mathbf{j} + 3\mathbf{k}$, $\mathbf{v} = -2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}$, and $\mathbf{w}

1. **State the problem:** We want to prove that three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ are coplanar, meaning they lie in the same plane. 2. **Formula and rule:** Three v

1. **State the problem:** Prove that $$\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) + \mathbf{v} \cdot (\mathbf{w} \times \mathbf{u}) + \mathbf{w} \cdot (\mathbf{u} \times \math

1. **Problem Statement:** Find the volume of the parallelepiped formed by the vectors \( \mathbf{u} = 3\mathbf{i} + 2\mathbf{k} \), \( \mathbf{v} = \mathbf{i} + 2\mathbf{j} + \math

1. **Problem Statement:** We have a parallelogram OACB with vectors \(\vec{OA} = \vec{a}\) and \(\vec{OB} = \vec{b}\).