Cross Product using Determinant Method
Given vectors \(\mathbf{a} = 2\mathbf{i} – 3\mathbf{j} + \mathbf{k}\) and \(\mathbf{b} = -\mathbf{i} + 4\mathbf{j} – 2\mathbf{k}\), calculate the cross product \(\mathbf{a} \times \mathbf{b}\) using the determinant method.
Manipulating Vectors in i, j, k Notation
Given the vectors \(\vec{v} = 3i – 4j + k\) and \(\vec{w} = 5i + 2j – 3k\), simplify the expression \(2\vec{v} – \vec{w}\) and express the result in the form \(ai + bj + ck\).
Vector Notation in 2D and 3D
This question examines your understanding of vector representation in two and three dimensions. Answer the following parts:
Part a
Given the 2D vector \(\vec{v} = 3\mathbf{i} + 4\mathbf{j}\), express this vector in component form \((x, y)\) and calculate its magnitude.
Part b
Consider the 2D vector \(\vec{v} = 3\mathbf{i} + 4\mathbf{j}\) extended into 3D by adding a z-component of 5, resulting in \(\vec{v} = 3\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}\). Express this 3D vector in component form and compute its magnitude.