1. The problem is to find the vector projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\) without using the orthogonal projection method. 2. The formula for the projection of \(\mathbf{a}\) onto \(\mathbf{b}\) is given by: $$\text{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}$$ where \(\cdot\) denotes the dot product. 3. To solve this, first compute the dot product \(\mathbf{a} \cdot \mathbf{b}\). 4. Then compute the dot product \(\mathbf{b} \cdot \mathbf{b}\), which is the square of the magnitude of \(\mathbf{b}\). 5. Divide the first dot product by the second to get the scalar multiplier. 6. Multiply this scalar by vector \(\mathbf{b}\) to get the projection vector. 7. This method does not require finding the orthogonal component explicitly and directly gives the projection vector. This completes the solution for 3b without using orthogonal projection.