📘 vector algebra

1. **Problem:** Calculate the dot product $\vec{u} \cdot \vec{v}$ and classify the angle between $\vec{u}$ and $\vec{v}$ for the vectors: (a) $\vec{u} = (2,8)$ and $\vec{v} = (-3,1

1. **Problem 1:** Given triangle OAB with vectors \(\vec{OA} = 5\vec{a}\) and \(\vec{OB} = 2\vec{b}\), point T lies on AB such that \(AT : TB = 5 : 1\). Show that \(\vec{OT}\) is p

1. **Problem statement:** In pentagon OABCD, given that OA is parallel to DC, AB is parallel to OD, with OD = 2AB and OA = 2DC, and vectors \(\overrightarrow{OA} = \mathbf{a}\) and

1. **Problem statement:** Calculate the angle between vectors $\mathbf{a} = (1,2)$ and $\mathbf{c} = (-2,5)$ in degrees. 2. **Formula:** The angle $\theta$ between two vectors $\ma

1. The problem is to calculate the angle between the vectors $\vec{a} = (1, 2)$ and $\vec{c} = (-2, 5)$. 2. The formula to find the angle $\theta$ between two vectors $\vec{u}$ and

1. **State the problem:** We are given vectors \(\overline{A} = (4, -6)\), \(\overline{B} = (1, 4)\), and \(\overline{P} = (-1, -15)\). We want to verify or find \(\overline{P}\) u

1. **State the problem:** We have two unit vectors $\mathbf{A}$ and $\mathbf{B}$ such that their sum is $\mathbf{A} + \mathbf{B} = 1$. We want to find the dot product $\mathbf{A} \

1. **State the problem:** We need to find the sum of two vectors $\mathbf{s}$ and $\mathbf{t}$, where $\mathbf{s} = \langle 1, -4 \rangle$ and $\mathbf{t} = \langle -2, 5 \rangle$.

1. **State the problem:** Calculate the dot product $a \cdot b$ for the given vectors. 2. **Recall the dot product formula:** For vectors $a = \langle a_1, a_2 \rangle$ and $b = \l

1. **Problem statement:** Find two unit vectors orthogonal to the given vectors $\mathbf{a}$ and $\mathbf{b}$. 2. **Formula and concept:** Two vectors are orthogonal if their dot p

1. **State the problem:** We need to compute the cross product $\mathbf{a} \times \mathbf{b}$ where $\mathbf{a} = \langle 3, 0, -1 \rangle$ and $\mathbf{b} = \langle 1, 2, 2 \rangl

1. **State the problem:** Find the vector sum $\vec{AB} + \vec{BC} + \vec{CA}$ where $A$, $B$, and $C$ are vertices of a triangle.

1. **Problem 25:** Find a unit vector in the same direction as the vector $\langle 8, -1, 4 \rangle$. 2. **Formula:** A unit vector $\mathbf{u}$ in the direction of vector $\mathbf

1. **Stating the problem:** In the figure, the tip of vector $\mathbf{c}$ and the tail of vector $\mathbf{d}$ are both the midpoint of segment $QR$. We need to express $\mathbf{c}$

1. **Problem statement:** Given the vectors in the quadrilateral with points A, B, C, D, and the relations AB = DC, DA = CB, DE = EB, EA = CE, write each sum or difference as a sin

1. El problema pide determinar el módulo y la dirección de las operaciones vectoriales dadas, y graficarlas. 2. Primero, recordemos que el módulo de un vector $\vec{A} = (x,y)$ se

1. **Stating the problem:** We are given that $\overrightarrow{AB} = \mathbf{p}$.

1. **Stating the problem:** We have an equilateral triangle ABC with side length BC = 1 unit.

1. The problem states that $P$ and $R$ are coplanar vectors and defines $X = P - R$. 2. To find the vector $X$, recall the vector subtraction rule: $$X = P - R = P + (-R)$$ where $

1. Let's prove the first identity: $$(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{a}

1. **Problem statement:** Find the vector perpendicular (orthogonal) to the vector $\vec{v} = (3, -4)$ in $\mathbb{R}^2$. 2. **Formula and concept:** In $\mathbb{R}^2$, a vector pe