Vector Algebra Problems | Mathix (MathGPT)

Vector Expression

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

Unit Vector Orthogonal

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

Parallelogram Area

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

Vector Pq Ob

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

Cross Product

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 \

Vector Midpoint Ratio

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

Cross Product Sine Rule

1. **Problem 1:** Find the geometrical interpretation of the cross product of two vectors and prove the sine rule in triangle ABC using vector methods. 2. **Geometrical interpretat

Vector Volume

1. Розглянемо умову $\vec{a} \times \vec{b} = \vec{0}$. Це означає, що вектори $\vec{a}$ і $\vec{b}$ колінеарні. 2. Вектори $\vec{a} = (5; 3; x)$ і $\vec{b} = (y; 6; 4)$

Vector Components Cube

1. **Calculate the scalar products:** Given vectors $\overrightarrow{BA}, \overrightarrow{BC}, \overrightarrow{CA}, \overrightarrow{DC}, \overrightarrow{AB}$, we first need their

Vector Path 3D

1. **Problem Statement:** We are given points in 3D space:

Vector Ac

1. **State the problem:** We are given quadrilateral OACB with vectors $ \vec{OA} = 4a $, $ \vec{OB} = 3b $, and $ \vec{BC} = 2a + b $. We need to find the vector \( \vec{AC}

Scalar Triple

1. **State the problem:** Compute the scalar triple product $[\hat{i},\hat{j},\hat{k}] \cdot [\hat{i} - \hat{j} + \hat{k}]$. 2. **Given vectors:** Let $\mathbf{a} = [\hat{i}, \hat{

Vector Extension

1. **State the problem:** Given a parallelogram OABC with $\overrightarrow{OA} = 3a$ and $\overrightarrow{OB} = 4b$, points C, B, and X are collinear with $CB : BX = 6 : 1$.

Vector Value

1. সমস্যা: আমরা দুটি ভেক্টর দিয়েছি $\vec{A} = 5\hat{i} + 2\hat{j} - 3\hat{k}$ এবং $\vec{B} = 15\hat{i} + a\hat{j} - 9\hat{k}$। এখানে $a$ এর মান নির্ণয় করতে হবে। 2. সমাধান: \nযদি

Vector Scalar Multiple

1. **State the problem:** We are given two vectors $\vec{A} = 5\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}$ and

Vector Operations

1. **Stating the problem:** We have vectors $\mathbf{a} = \langle -2, 4 \rangle$, $\mathbf{b} = \langle 2, -2 \rangle$, $\mathbf{c} = \langle 4, 6 \rangle$.

Vector Cross Sum

1. The problem states: If $\vec{A}$ is a non-zero vector, we want to find the value of: $$\|\vec{A} \times \hat{i}\|^2 + \|\vec{A} \times \hat{j}\|^2 + \|\vec{A} \times \hat{k}\|

Vector Products

1. **State the problem:** Given vectors $\vec{M} = 2i - 7j + 4k$ and $\vec{N} = 3i - 5j + k$, find:

Vector An

1. **State the problem:** Given vectors $\overrightarrow{PA} = \begin{pmatrix} -6 \\ -8 \\ -6 \end{pmatrix}$ and \(\overrightarrow{PN} = \begin{pmatrix} 6 \\ 2 \\ -6 \end{pmatrix

Angle Vector

1. **Problem statement**: Given points A and B with position vectors $\mathbf{A} = \mathbf{i} + 7\mathbf{j} + 2\mathbf{k}$ and \(\mathbf{B} = -5\mathbf{i} + 5\mathbf{j} + 6\mathb

Vector Operations

1. **State the problem:** Given position vectors relative to origin $O$: