Vector Addition
Given the vectors \(\vec{a} = 3\hat{i} - 2\hat{j} + 4\hat{k}\) and \(\vec{b} = -\hat{i} + 2\hat{j} + 2\hat{k}\), find the result of the vector addition \(\vec{a} + \vec{b}\).
Vector Subtraction
Given two vectors \(\vec{a} = 3\hat{i} - 2\hat{j} + \hat{k}\) and \(\vec{b} = -\hat{i} + 2\hat{j} - 3\hat{k}\), find the vector \(\vec{a} - \vec{b}\).
Vector Multiplication by a Scalar
If a vector \(\vec{v} = 5\vec{i} - 3\vec{j} + \vec{k}\), what is the result of the operation \(7\vec{v}\)?
Finding the Variables
Given that the vector \( \mathbf{a} = x \mathbf{i} + 2 \mathbf{j} \) and the vector \( \mathbf{b} = 3 \mathbf{i} + y \mathbf{j} \) are orthogonal, find the values of variables \(x\) and \(y\).
Finding the Length
Find the length of the vector \( \vec{v} = 3\hat{i} - 2\hat{j} + 4\hat{k} \)
Determining Column Spaces
Given the following vectors for the matrix A: \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \), what is the column space of A?
Finding the Angle Between the Vectors
Given vectors \( \mathbf{A} = 3\mathbf{i} - 2\mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k} \), find the angle between these two vectors.