Vector Addition
Given vector \(\mathbf{a} = [3, 2, 1]\) and vector \(\mathbf{b} = [1, 4, 2]\), find the sum of these two vectors \(\mathbf{a} + \mathbf{b}\).
Vector Subtraction
Let \(\vec{a} = 3\hat{i} - 4\hat{j} + 2\hat{k}\) and \(\vec{b} = -\hat{i} + 2\hat{j} - 3\hat{k}\). Find \(\vec{a} - \vec{b}\).
Vector Multiplication by a Scalar
Let vector \( \mathbf{v} = [4, -3, 2] \). Find the result of \( -2 \mathbf{v} \).
Finding the Variables
Given the vectors \( \vec{A} = 2\vec{i} - 3\vec{j} + k \) and \( \vec{B} = -\vec{i} + 2\vec{j} + 4k \). If \( \vec{A} \) and \( \vec{B} \) are orthogonal to each other, find the value of \( k \).
Finding the Length
Find the length of the vector \(v = (4, 3, 12)\).
Finding the Position Vector
Given two points A (1, 2, 3) and B (4, -1, 5), find the position vector of B in respect to A.
Determining Column Spaces
Determine the column space for the matrix \[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
Finding an Orthonormal Basis by Gram-Schmidt Method
Given a set of vectors \(V = \{v_1, v_2, v_3\}\) in \(\mathbb{R}^3\) with \(v_1 = [1, 0, 0]\), \(v_2 = [1, 1, 0]\), and \(v_3 = [1, 1, 1]\), find an orthonormal basis for this set using the Gram-Schmidt process.