📘 vector algebra

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

1. **Тодорхойлолт:** 𝐴𝐵𝐶 гурвалжны 𝐴 оройгоос татсан биссектрисийн суурь нь 𝑃 ба 𝑄 нь 𝐴𝐵 талын дундаж цэг байна. 𝑃𝑄⃗⃗⃗⃗⃗⃗ векторыг 𝑏⃗⃗, 𝑐⃗ векторуудаар илэрхийлэх шаардлагатай. 2.

1. Let's start by stating the problem: You want to understand why the cross product of two vectors is not equal to $|a|\sin t + ab\cos t + a^2$. 2. The cross product of two vectors

1. The problem involves the vector expression $$\vec{IIHA} + \vec{MBII}$$ where arrows indicate these are vectors. 2. To add vectors, we use the rule: $$\vec{A} + \vec{B} = \vec{C}

1. **Stating the problem:** We have a point A at the origin with three ropes extending along the positive y-axis to B (1.5 m), negative x-axis to C (2 m), and positive z-axis to D

1. **State the problem:** We need to find the equation of a plane that passes through the point $P(2,1,-3)$ and contains the vectors $\vec{v_1} = 3\mathbf{i} + \mathbf{j} + 4\mathb

1. **Problem statement:** Prove that the diagonals of a parallelogram bisect each other using vectors, without assuming the origin. 2. **Setup:** Let the parallelogram have vertice

1. **State the problem:** Express the vector $\mathbf{v} = 5\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$ as a sum of two vectors, one parallel and one perpendicular to $\mathbf{u} = 2\m

1. **Problem statement:** Find the value of $k$ such that $\overrightarrow{AP} = \frac{3}{2} \overrightarrow{PB}$. 2. **Recall vector relation:** Since $P$ lies on line $AB$, we ha

1. **State the problem:** We need to find a vector equation for the line $l_1$ passing through points $A(2,5,9)$ and $B(6,0,10)$. 2. **Formula for vector equation of a line:** The

1. Problem: Find the value of $|3\mathbf{v} + \mathbf{w}|$ where $\mathbf{v} = 3\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}$ and $\mathbf{w} = 5\mathbf{i} - \mathbf{j} + 3\mathbf{k}$. 2

1. **Problem Statement:** Find a unit vector perpendicular to the plane formed by vectors \(\vec{A} = 2\mathbf{i} - 3\mathbf{j} - \mathbf{k}\) and \(\vec{B} = \mathbf{i} + 4\mathbf

1. **Problem statement:** Find the unit vector $\vec{C}$ that makes an angle of 60° with $2\hat{i} + 2\hat{j} - \hat{k}$ and an angle of 45° with $\hat{i} - \hat{k}$. Then compute