📘 vector algebra

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

1. Problem: Given parallelogram OABC with origin at O, vectors \(\vec{OA} = \mathbf{p}\) and \(\vec{OC} = \mathbf{q}\). M is midpoint of OB, N divides AB in ratio 3:2. Find vectors

1. **Stating the problem:** We need to find the position vector of point D, where lines CB and ON are extended to meet, expressed in terms of vectors $p$ and $q$. 2. **Analyzing th

1. The problem: Given 5 vectors, find their resultant vector such that it can land in each of the four quadrants. 2. We first remember that the resultant vector ${\vec R}$ is the v

1. Let's start by understanding the problem: We need to create 5 vectors such that their resultant vector lands in each of the 4 quadrants of the coordinate plane. 2. Recall that t

1. Stating the problem: We want to create 5 vectors such that their resultant vector lies in each of the four quadrants of the coordinate plane. 2. Understanding vectors and quadra

1. **Problem Statement:** Given three non-coplanar vectors $a$, $b$, and $c$, express the cross products $b \times c$, $c \times a$, and $a \times b$ in terms of $a$, $b$, and $c$.

1. **State the problem:** We are given vectors \( \overrightarrow{a} = -2\overrightarrow{i} - n\overrightarrow{j} \), \( \overrightarrow{b} = n\overrightarrow{j} \), and a unit vec

1. Diberi bahawa $ST = \mathbf{a} = -2\mathbf{i} - n\mathbf{j}$ dan $PQ = 2\mathbf{b} = 2n\mathbf{j}$ kerana $\mathbf{b} = n\mathbf{j}$. 2. Berdasarkan segi empat selari $PQRS$, ki

1. Stating the problem: We have vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{b}\) displayed as shown, and we want to find which vector operation matches \(\vec{b}\) in terms of \(\

1. **Stating the problem:** Given vectors ū (vertical up), v (left-down), and b (up-right), identify which operation corresponds to the vector shown in the image. 2. **Analyzing ve

1. **State the problem:** We need to find the unit vector and direction of the vector $\mathbf{A} = 3\mathbf{i} - 7\mathbf{j} + 4\mathbf{k}$.\n\n2. **Calculate the magnitude of vec

1. **State the problem:** We need to find the unit vector and direction of a given vector $\mathbf{v}$. The direction is the angle the vector makes with the positive x-axis. 2. **F