📘 vector algebra

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

1. **Problem statement:** Given triangle OAB with position vectors of points A and B from O as $\mathbf{a}$ and $\mathbf{b}$ respectively. Given ratios: $OC : CA = \frac{2}{3}$ and

1. **State the problem:** Find the angle between the two lines given by the parametric equations: Line 1: $x = 9 - 6t$, $y = -5t - 10$, $z = 0$

1. **Top-left: Find parametric equation of line through P(-8,5,3) parallel to xy and xz planes.** - A line parallel to the xy-plane has constant $z$.

1. **State the problem:** Find the angle between two lines given in various forms (parametric, vector, symmetric). 2. **Formula:** The angle $\theta$ between two lines with directi

1. **Problem Statement:** Find the angle between the two lines given by the parametric equations: $$x=9-6t,\quad y=-5t-10,\quad z=0$$

1. مسئله: مقدار $m$ را بیابید به طوری که دو بردار $$\vec{b} = (m-1)\vec{i} - \vec{j} + \vec{k}$$

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