📘 vector algebra

1. **Problem Statement:** Find the angle each vector makes with the positive x-axis. 2. **Formula:** The angle $\theta$ a vector $\vec{v} = ai + bj$ makes with the positive x-axis

1. **Problem statement:** Given a regular hexagon centered at point $O$, with vectors $\overrightarrow{AB} = 3p + q$ and $\overrightarrow{BC} = 4p$, find:

1. **State the problem:** We have triangle ABC with M as the midpoint of AC. Given vectors \(\overrightarrow{AB} = 8\mathbf{a} - 4\mathbf{b}\) and \(\overrightarrow{BC} = 10\mathbf

1. **State the problem:** We are given vectors \(\overrightarrow{OA} = 5a + 8b\) and \(\overrightarrow{OB} = 6a - b\). We need to find the vector \(\overrightarrow{AB}\) in terms o

1. The problem asks to find the magnitude of the vector $\mathbf{v} = 3\mathbf{i} - 4\mathbf{j} + 12\mathbf{k}$.\n\n2. The magnitude (or length) of a vector $\mathbf{v} = a\mathbf{

1. **Problem:** A courier company has a central depot at point O. A delivery motorbike follows the route: Depot (O) → Address A: $\vec{v}_1 = 8i + 10j$ km; A → B: $\vec{v}_2 = 7i +

1. **Problem statement:** Given three non-zero vectors $\vec{A}, \vec{B}, \vec{C}$ such that $\vec{A} + \vec{B} = \vec{C}$, with $\|\vec{A}\| = \|\vec{B}\|$ and $\|\vec{C}\| = \sqr

1. **State the problem:** Find the vector equation of the line given the symmetric equations:

1. **Stating the problem:** Given points Tink M $(2, -7)$ and Oleh $(3, 4)$, and a vector equation $8x + 3y = 9$, find the projection of the vector from Tink M to Oleh onto the vec

1. **Problem Statement:** Prove that for any triangle ABC with usual notations, the vector magnitude relation holds: $$|\vec{a}| = |\vec{b}| \cos \gamma + |\vec{c}| \cos \beta$$

1. **Problem:** Find the position vector of vertex D of parallelogram ABCD given position vectors of A, B, and C. 2. **Given:**

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

1. **Problem Statement:** Verify if the given information about lines L and M is correct based on the provided direction ratios, parametric equations, and symmetric equations. 2. *

1. **Stating the problem:** We have a parallelogram ABCD with vectors \(\vec{DA} = \vec{a}\) and \(\vec{DC} = \vec{c}\). Points M and N are midpoints of segments CB and AB respecti

1. The problem asks why a vector \(\vec{A}\) has an angle of 270°. 2. In a 2D coordinate system, angles are typically measured from the positive x-axis, moving counterclockwise.

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 :