Vector Algebra Problems

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 $O$:

Vector Operations

1. **Problem Statement:** (a) Find the magnitude and direction of the displacement vector $\overrightarrow{AB}$ between points $A(3, 5)$ and $B(6, -4)$.

Vector Problems

1. Problem: Given vector $v = (-1, 2, 5)$, find all scalars $k$ such that $||kv|| = 4$. Step 1: Recall that $||kv|| = |k| imes ||v||$.

Vector Projection

1. Uppgáva 8: Finn krosstølini fyri projektiðina av vektaranum $\vec{a} = \begin{pmatrix}30\\-10\end{pmatrix}$ á $\vec{b} = \begin{pmatrix}4\\3\end{pmatrix}$. 2. Projektilin av

Vector Concepts

1. The problem is to solve a given unspecified problem using vector concepts. 2. To proceed, you need to specify the vectors involved or the exact problem statement (such as vector

Vector Orthogonality

1. Xét khẳng định b) $\overrightarrow{AM} = (1 - k)\overrightarrow{AB} + k\overrightarrow{AC}$ và c) \(\overrightarrow{PN} = -\frac{4}{15}\overrightarrow{AB} + \frac{1}{3}\overri

Vector Perpendicularity

1. **Problem:** Find $k$ such that $\vec{AM} \perp \vec{PN}$ given $$\vec{AM}=(1-k)\vec{AB} + k\vec{AC}, \quad \vec{PN} = - \frac{4}{15} \vec{AB} + \frac{1}{3} \vec{AC}$$ and provi