📘 vector algebra

1. **State the problem:** We are given points $A(-1,2)$, $B(5,-2)$, $C(1,3)$, and $D(0,0)$. We want to express the vector $\overrightarrow{BA}$ as a linear combination of vectors $

1. Planteamos el problema: En un hexágono regular, se nos da el vector \(\overrightarrow{CX} = -3\mathbf{u} + 2\mathbf{v} + \frac{3}{2} \mathbf{w}\) y queremos hallar el punto \(X\

1. **Stating the problem:** We want to understand why the condition for a vector $w$ to be equidistant from vectors $u$ and $v$ is given by

1. **State the problem:** We are given four vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ with the sum $\mathbf{a} + \mathbf{b} + \mathbf{c} + \mathbf{d} = \la

1. **State the problem:** Convert the vector $\mathbf{A} = 3\mathbf{a}_x + 4\mathbf{a}_y$ into cylindrical components at $\theta = 53.1^\circ$ and find the radial component $A_r$.

1. **Problem statement:** In parallelogram ABCD, point M is the midpoint of side BC. Given vectors $\overrightarrow{AB} = \vec{a}$ and $\overrightarrow{AD} = \vec{b}$, find vectors

1. **Problem statement:** In parallelogram ABCD, point M is the midpoint of side BC. Given vectors \(\overline{AB} = \vec{a}\) and \(\overline{AD} = \vec{b}\), express vectors \(\o

1. **State the problem:** Find the direction angle $\theta$ of the vector $\vec{v} = (-3, -10)$ measured from the positive x-axis. 2. **Formula:** The direction angle $\theta$ is g

1. **State the problem:** Given vectors $a$ and $b$ with magnitudes $|a|=15$, $|b|=20$, and $|a-b|=15.5$, find the magnitude $|a+b|$. 2. **Recall the formula for the magnitude of t

1. **State the problem:** Find the vector equation of a line parallel to the line given by

1. **State the problem:** We are given position vectors of two ships A and B as functions of time $t$ hours:

1. **Problem statement:** Given that $\mathbf{a}$ and $\mathbf{c}$ are unit vectors, the magnitude of $\mathbf{b}$ is 4, and the angle between $\mathbf{b}$ and $\mathbf{c}$ is 60 d

1. **State the problem:** We are given vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ such that $\mathbf{a} \neq \mathbf{0}$ and $$\mathbf{a} \times 3\mathbf{b} = 2 \mathbf{a

1. **State the problem:** Given vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ with $\mathbf{a} \neq \mathbf{0}$, and the equation $\mathbf{a} \times 3\mathbf{b} = 2 \mathbf{

1. **Problem statement:** Given that the cross product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is zero, i.e., $\mathbf{u} \times \mathbf{v} = \mathbf{0}$, what can we deduce a

1. **Problem:** Use the parallelogram method to add vectors $\vec{x} = \langle 3, 4 \rangle$ and $\vec{y} = \langle 4, 1 \rangle$. 2. **Formula:** Vector addition by the parallelog

1. Let's start by stating the problem: Understanding how vector calculations work and elaborating on the basic operations. 2. Vectors are quantities with both magnitude and directi

1. **State the problem:** Given points $A(-2,0,10)$, $B(1,9,3)$, and $C(2,-4,6)$, find vectors $\overrightarrow{AC}$ and $\overrightarrow{BC}$, then find the angle $\angle ACB$. 2.

1. **Problem:** Calculate the dot product (also called scalar product) of the vectors given in part 1a: $\mathbf{u} = (2, -3)$ and $\mathbf{v} = (1, 2)$. 2. **Formula:** The dot pr

1. **State the problem:** Find a unit vector perpendicular to both vectors $\mathbf{a} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{b} = -2\mathbf{i} + 3\mathbf{j} + 2\mat

1. **State the problem:** Find a unit vector perpendicular to both vectors $\mathbf{a} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{b} = -2\mathbf{i} + 3\mathbf{j} + 2\mat