📘 vector algebra

1. **State the problem:** We are given vectors $\mathbf{a} = 2\mathbf{i} + 5\mathbf{j}$, $\mathbf{b} = 12\mathbf{i} - 10\mathbf{j}$, and $\mathbf{c} = -3\mathbf{i} + 9\mathbf{j}$.

1. **Problem:** Compute the dot product $\vec{a} \cdot \vec{b}$ for given vectors. **Formula:** The dot product of two vectors $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b

1. **State the problem:** We need to find which vectors from the given list are parallel to the vector $$\overrightarrow{FG} = 2a - 3b$$ where $$a$$ and $$b$$ are non-parallel vect

1. **Problem Statement:** We are given vectors $a$ and $b$ which are not parallel, and a vector $\overrightarrow{XY} = 2a - 5b$. We need to determine which of the given vectors are

1. **Problem Statement:** We are given vectors $\mathbf{a}$ and $\mathbf{b}$ which are not parallel, and a vector $\overrightarrow{XY} = 2\mathbf{a} - 5\mathbf{b}$. We need to dete

1. **Problem statement:** We have triangle ABC with M as the midpoint of line segment AC. Given vectors:

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

1. **Problem Statement:** Given parallelogram ABCD with points L and M as midpoints of sides BC and CD respectively, prove that $$\vec{AL} + \vec{AM} = \frac{3}{2} \vec{AC}$$.

1. **State the problem:** We are given vectors \(\overrightarrow{AB} = \begin{pmatrix} 9 \\ -3 \end{pmatrix}\) and \(\overrightarrow{AC} = \begin{pmatrix} 18 \\ -12 \end{pmatrix}\)

1. **Problem Statement:** We are given a point $P$ with position vector

1. **Problem Statement:** We have points A and B with position vectors \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\) and \(\overrightarrow{OB} = 2\mathbf{i} - 3\

1. **Problem Statement:** (i) Show that points with position vectors $\mathbf{A} = \mathbf{i} - \mathbf{j}$, $\mathbf{B} = 4\mathbf{i} + 3\mathbf{j} + \mathbf{k}$, and $\mathbf{C}

1. **Problem Statement:** We are given fundamental definitions related to vectors in space, including unit vectors, equal vectors, zero vectors, negative vectors, scalar multiplica

1. **Problem Statement:** We want to understand the components of a vector and how to find the magnitude of a vector in 3D space. 2. **Vector Components:** A vector \( \mathbf{r} \

1. **Problem statement:** We have two vectors \( \vec{a} \) and \( \vec{b} \) each with magnitude 8 units. \( \vec{a} \) makes a 45° angle with the positive x-axis, and \( \vec{b}

1. **Problem Statement:** Understand the fundamental concepts and properties of vectors in three-dimensional space $\mathbb{R}^3$ including unit vectors, components, magnitude, equ

1. **Problem Statement:** Find the position vector of a point dividing a line segment in a given ratio, and prove some vector geometry theorems. 2. **Formula for internal division:

1. **Problem:** Find the position vector of point Q dividing the line segment \(\overline{AB}\) externally in the ratio 3 : 2. 2. **Formula:** If a point Q divides the line segment

1. **Problem Statement:** Given points P = (3, -1), Q = (-4, -6), R = (1, 4), and S = (2, 5), find the following vectors in component form: (i) Vector $\overrightarrow{PQ}$

1. **Problem Statement:** We are given the equation $$R \sin \beta = P \sin \gamma + Q \sin \alpha$$ and the substitutions $$\sin \beta = \frac{r}{om}, \quad \sin \gamma = \frac{p}

1. Let's clarify the problem: You mentioned that the answer is $x - y$ equals the vector $\mathbf{FA}$. We need to understand what $x$, $y$, and $\mathbf{FA}$ represent in this con