Simple Dot Product Calculation
Calculate the dot product of the vectors \(\vec{a} = (3, -2)\) and \(\vec{b} = (4, 5)\).
Reviewing 2D Vector Magnitude Calculation
Find the magnitude of the vector \(\vec{v} = \langle 3, 4 \rangle\).
Part a
In the 2D case, let vector \(\mathbf{a} = (3, 4)\) and vector \(\mathbf{b} = (2, 0)\). Compute the projection of \(\mathbf{a}\) onto \(\mathbf{b}\).
Dot Product Computation in 2D and 3D
Given vectors \(\vec{a} = (2, -1, 3)\) and \(\vec{b} = (-1, 4, 0)\), compute the dot product \(\vec{a}\cdot\vec{b}\). Also, given vectors \(\vec{u} = (3, 4)\) and \(\vec{v} = (2, -5)\), compute their dot product.
2D Vector Addition of Two Vectors
Given two vectors \(\vec{A} = (3, 4)\) and \(\vec{B} = (-2, 5)\), find the resultant vector \(\vec{A} + \vec{B}\).
Part a
For the two-dimensional vector \(\mathbf{v} = [4, -3]\): 1. Determine its direction angle \(\theta\) relative to the positive x-axis. Use \(\theta = \arctan\left(\frac{y}{x}\right)\) and adjust the angle to be in the correct quadrant. 2. Compute its unit vector \(\hat{\mathbf{v}}\).
Normalization of a 2D Vector
Given the vector \(\vec{v} = \langle 3, 4 \rangle\), find the unit vector in the direction of \(\vec{v}\).
Vector Notation in 2D and 3D
Explain how a vector represented in 2D as \( (x, y) \) is extended to 3D as \( (x, y, z) \), and discuss the implications for vector operations such as addition and scalar multiplication.