📘 vector algebra

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

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}

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{

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

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

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

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

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}\|

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

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

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

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

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

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

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

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

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

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

**Problem Statement:** We have multiple vector equations and properties related to a parallelogram, triangle, square, and trapezoid. We will analyze and verify the vector equalitie

1. The problem gives two vectors $\mathbf{a} = 11 \mathbf{i} + 9 \mathbf{j} + 0 \mathbf{k}$ and $\mathbf{b} = x \mathbf{i} + 7 \mathbf{j} + 0 \mathbf{k}$. 2. Find $x$ such that $\m

1. The problem is to draw and understand the vectors $\overrightarrow{VAB}$ and $\overrightarrow{VAD}$.\n\n2. Generally, $\overrightarrow{VAB}$ represents the vector from point A t