📘 vector algebra

1. **State the problem:** We have a parallelogram OABC with vectors $\overrightarrow{OA} = 14a$ and $\overrightarrow{OB} = 7b$. Point D lies on segment AC such that the ratio $AD :

1. **State the problem:** We need to find the vector $\vec{AD}$ in terms of $a$ and $b$ given $\vec{OA} = 8a$, $\vec{OB} = 4b$, $C$ is midpoint of $AB$, and $D$ lies on $OC$ with r

1. **Problem Statement:** We are given a parallelogram ABCD with vectors $\vec{a}$ and $\vec{b}$ representing sides $\overrightarrow{AB}$ and $\overrightarrow{AD}$ respectively.

1. The problem is to understand what happens when multiplying horizontal directions. 2. In mathematics and physics, directions are often represented as vectors.

1. **State the problem:** We need to determine if the ordered triple of vectors $(\mathbf{c}, \mathbf{a}, \mathbf{b})$ forms a left hand triple, right hand triple, or neither. 2. *

1. **State the problem:** We need to determine if the ordered triple of vectors $(\mathbf{c}, \mathbf{a}, \mathbf{b})$ forms a left hand triple, right hand triple, or neither. 2. *

1. **Problem Statement:** We need to find the point to which the vector $\frac{1}{4} \overrightarrow{OA} + \frac{3}{4} \overrightarrow{OB}$ points. 2. **Understanding the vectors:*

1. **Problem statement:** Find the angle between the vectors $$3\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}$$ and $$-4\mathbf{i} - \mathbf{j} + 3\mathbf{k}$$. 2. **Formula used:** The a

1. **State the problem:** Determine which pairs of lines are skew lines. Skew lines are lines that are not parallel and do not intersect. 2. **Recall definitions and formulas:**

1. **State the problem:** We have two vectors \( \vec{G} \) and \( \vec{H} \) in 3D space with magnitudes and angles given. We need to find:

1. **Problem statement:** We have a regular hexagon centered at the origin O.

1. **Problem statement:** Given that $\overrightarrow{AB} = 2a + 18b$ and point $P$ lies on line $AB$ such that the ratio $AP : PB = 1 : 3$, find the vector $\overrightarrow{AP}$ i

1. **State the problem:** Given vector $\overrightarrow{AB} = -7\mathbf{i} + 3\mathbf{j} + 8\mathbf{k}$ and point $A(-2, -3, 6)$, find the coordinates of point $B$. 2. **Recall the

1. **State the problem:** Find a vector of magnitude 7 in the direction of \( \mathbf{v} = 12\mathbf{i} - 5\mathbf{k} \). 2. **Formula and rules:** To find a vector of a given magn

1. **Problem statement:** Given vectors $a$, $b$, and $c$ such that $a \neq 0$ and $a \times 3b = 2a \times c$, show that $3b - 2c = \lambda a$ for some scalar $\lambda$. 2. **Reca

1. **Stating the problem:** We are given a rectangular prism (balok) with vertices labeled as follows:

1. We check each exercise for correctness by verifying the vector calculations and parameter substitutions. 2. For the first exercise (points on line AB):

1. The problem asks to find the vector $\vec{AB}$ given points $A(-2,6)$ and $B(8,-5)$.\n\n2. The formula for vector $\vec{AB}$ is $\vec{AB} = (x_B - x_A)\hat{i} + (y_B - y_A)\hat{

1. **Problem statement:** Given two vectors in 3D space $\vec{u} = (2, -1, 3)$ and $\vec{v} = (1, 2, -2)$, find the coordinates of the vector $\vec{w} = [\vec{u}, \vec{v}]$, which

1. **Problem:** Given two vectors $\vec{a} = (1, 2, -1)$ and $\vec{b} = (3, 0, 2)$ in space $Oxyz$, find the coordinates of the cross product vector $[\vec{a}, \vec{b}]$. 2. **Form

1. **Stating the problem:** We are given two displacement vectors for a drone's journey: \(\overrightarrow{AB} = (4,8,6)\) km and \(\overrightarrow{BC} = (-1,-4,6)\) km. We need to