📘 vector algebra

1. **Problem statement:** Given a regular hexagon ABCDEF with center O, vectors AB = $\vec{x}$ and BC = $\vec{y}$, express vectors $\vec{ED}$, $\vec{DE}$, $\vec{FE}$, $\vec{AC}$, $

1. **Problem statement:** Find the area, sine, and cosine of each angle of the triangle with vertices A(0,0,0), B(4,-1,3), and C(1,2,3). 2. **Step 1: Find vectors AB and AC.**

**Problem 9:** Find two unit vectors parallel to the yz-plane and orthogonal to $\mathbf{v} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}$. 1. Vectors parallel to the yz-plane have zero

1. Problem: Determine the magnitude of each vector given in the image. Since the image is not provided, we cannot calculate exact magnitudes here.

1. The problem asks to find the magnitude of the sum of two vectors $\tilde{q}$ and $\tilde{v}$, given that $\epsilon = |\tilde{q}|$, $1 = |\tilde{v}|$, and $\frac{\epsilon_1}{\eps

1. **State the problem:** Given vectors $a$ and $b$ with magnitudes $|a|=1$, $|b|=2$, and magnitudes of their sums and differences: $|a+b|=\sqrt{12}$, $|a-b|=\sqrt{10}$, and $|a+2b

1. **State the problem:** Given three non-zero vectors $a$, $b$, and $c$, where $c$ is a unit vector perpendicular to both $a$ and $b$, and the angle between $a$ and $b$ is $\frac{

1. **Problem statement:** We have a parallelogram OACB with vectors \(\vec{a} = \overrightarrow{OA}\) and \(\vec{b} = \overrightarrow{OB}\).

1. The question asks about vector principles, which are fundamental concepts in vector mathematics. 2. Vectors have both magnitude and direction, and they can be added, subtracted,

1. **Problem statement:** Given vectors $\mathbf{AB} = \mathbf{p}$, $\mathbf{CA} = \mathbf{q}$, and $\mathbf{DC} = 3\mathbf{AB} = 3\mathbf{p}$, we need to: (a) Express $\mathbf{DA}

1. **Problem statement:** Show that for any vector $\vec{v}$, the dot product $\vec{v} \cdot \vec{v} = |\vec{v}|^2$. 2. **Step 1:** Recall the definition of the dot product for a v

1. **State the problem:** We are given two vectors $\mathbf{a} = \mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$ and $\mathbf{b} = 6\mathbf{i} - \mathbf{j} + 3\mathbf{k}$. We need to find

1. **Problem statement:** Given trapezium OABC with AB \parallel OC, OC = 4 AB, D on OA with OD : DA = 3 : 1, E on OC with OE : EC = 1 : 3, express vectors ED and CB in terms of a

1. Problem: For vectors $\mathbf{p} = \mathbf{i} + 4\mathbf{j} - 3\mathbf{k}$ and $\mathbf{q} = 5\mathbf{i} - 2\mathbf{j}$, determine: (1) $\mathbf{p} \cdot \mathbf{q}$

1. **Problem statement:** Given that $a$ and $b$ are unit vectors, analyze the vector $v = a - b$ and determine which of the following statements is true: (a) $v$ is a zero vector.

1. The problem asks to analyze the expression $a \cdot (b \times a)$ and determine its nature. 2. Recall that $b \times a$ is the cross product of vectors $b$ and $a$, which result

1. **State the problem:** We need to find a unit vector $\mathbf{u}$ orthogonal to vectors $\mathbf{a} = \mathbf{i} + \mathbf{j} - \mathbf{k}$ and $\mathbf{b} = 2\mathbf{i} - \math

1. **Problem statement:** Given vectors $a$ and $b$ with magnitudes $\|a\| = \sqrt{3}$, $\|b\| = 2$, and the angle between them $\widehat{(a,b)} = 150^{0}$, find the area of the pa

1. **Problem 1: Find p and q given vector $\mathbf{a} = p\mathbf{i} + q\mathbf{j}$ with $|\mathbf{a}|=11$ and angle 20° with positive y-axis.** 2. The magnitude of $\mathbf{a}$ is

1. **State the problem:** We need to compute the cross product of vectors \(\mathbf{u} = \begin{bmatrix} 5 \\ -1 \\ -2 \end{bmatrix}\) and \(\mathbf{v} = \begin{bmatrix} -10 \\ 4 \

1. **State the problem:** We have triangle OAB with points P and Q defined as follows: