📘 vector algebra

1. **Show that the vectors $\vec{A} = 2\mathbf{i} - \mathbf{j} + \mathbf{k}$, $\vec{B} = \mathbf{i} - 3\mathbf{j} - 5\mathbf{k}$, and $\vec{C} = 3\mathbf{i} - 4\mathbf{j} - 4\mathb

1. **State the problem:** Find the equation of the line passing through points $A(4,1,2)$ and $B(6,-3,4)$.\n\n2. **Formula and rules:** The vector equation of a line through point

1. **State the problem:** Find the vector and Cartesian equations of the plane passing through points $A(-2,-2,2)$, $B(3,2,3)$, and $C(2,-2,2)$. 2. **Formula and rules:**

1. **State the problem:** Find the cross product $\vec{a} \times \vec{b}$ where $\vec{a} = 14\hat{i} + 13\hat{j} + 15\hat{k}$ and $\vec{b} = -9\hat{i} + 15\hat{j} + 11\hat{k}$.\n\n

1. ปัญหา: ทำไมเวกเตอร์ $\overrightarrow{BD}$ ถึงเท่ากับ $\mathbf{\bar{v}} = \overrightarrow{CB}$ ในรูปสี่เหลี่ยมด้านขนาน ABCD 2. กำหนดรูปสี่เหลี่ยมด้านขนาน ABCD:

1. **Problem:** Show that the vectors $\vec{A} = 2\mathbf{i} - \mathbf{j} + \mathbf{k}$, $\vec{B} = \mathbf{i} - 3\mathbf{j} - 5\mathbf{k}$, and $\vec{C} = 3\mathbf{i} - 4\mathbf{j

1. The problem asks to draw vectors with angles 120°, 240°, and 360° using the head-to-tail method and polygon representation. 2. Vectors can be represented graphically by arrows s

1. **Problem Statement:** Given vectors $\mathbf{a} = \overrightarrow{OA}$ and $\mathbf{b} = \overrightarrow{OB}$, point $P$ lies on $AB$ such that $BA = 4BP$, and $Q$ is the midpo

1. نبدأ ببيان المشكلة: لدينا متجه وحدة \( \vec{ء} \) عمودي على المستوى الذي يحوي المتجهين \( \vec{أ} \) و \( \vec{ب} \). 2. المعطى هو:

1. نبدأ بقراءة المسألة: لدينا متجه وحدة ى عمودي على المستوى الذي يحوي المتجهين أ و ب. 2. نعلم أن المتجه العمودي على المستوى الذي يحوي أ و ب هو متجه حاصل الضرب الاتجاهي \( (1+2\math

1. **Problem (a):** Find the distance between points $A(3,4,5)$ and $B(6,8,9)$. 2. **Formula:** The distance between two points in 3D is given by

1. **Find a vector of length 10 units along the line through A(0,−1,1) and B(2,−2,3):** Step 1: Find the vector \( \overrightarrow{AB} = B - A = (2-0, -2+1, 3-1) = (2, -1, 2) \).

1. **Stating the problem:** Given a triangle ABC with points E and F on sides AC and CB respectively, and points A, B, D collinear. Given ratios AE : AC = 2 : 3, CF : CB = 3 : 5, a

1. **Problem statement:** Given vectors $\vec{a} = (-5, 4, 1)$, $\vec{b} = (3, -5, -3)$, and $\vec{c} = (5, -2, 5)$, evaluate the expression $$\left(\frac{\vec{a} \cdot \vec{b}}{\v

1. **Problem statement:** Find the coordinates of point P that divides the line segment AB in the ratio 2:3, where A has coordinates $(3,-1,-4)$ and B has coordinates $(8,-6,6)$. 2

1. **State the problem:** We need to divide the vector $\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$ into three components, where one component is parallel to the vector $\vec{i}$. 2.

1. **State the problem:** We need to divide the vector $\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$ into three components such that one component is parallel to the vector $\vec{i}$.

1. **State the problem:** We are given a vector $\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$ and need to divide it into three components: one parallel to $\vec{i}$, one parallel to $\

1. **Problem statement:** Divide the vector $\vec{v} = 3\vec{i} + 5\vec{j} + \vec{k}$ into three components: one parallel to $\vec{i}$, one parallel to $\vec{i} + 2\vec{j} + 3\vec{

1. **Problem statement:** Given three vectors \(\vec{a}, \vec{b}, \vec{c}\) such that \(|\vec{a}|=2\), \(|\vec{b}|=3\), \(|\vec{c}|=4\), and \(\vec{a} + \vec{b} + \vec{c} = \vec{0}

1. **State the problem:** Given points $A(1,3)$ and $B(0,-1)$, construct the vector $\overrightarrow{AB}$ and find its magnitude and direction angle. 2. **Construct the vector $\ov