- Compute the vector sum by adding corresponding components: \(u + v = \langle -4 + 11, 7 - 6 \rangle = \langle 7, 1 \rangle\).
- Find the magnitude using the formula \(\sqrt{w_x^2 + w_y^2}\): \(\sqrt{7^2 + 1^2} = \sqrt{50} \approx 7.07\).
- The final results are \(u + v = \langle 7, 1 \rangle\) and \(\|u + v\| = 7.07 \text{ units}\).
- Final Answer: \(\boxed{u + v = \langle 7, 1 \rangle, \|u + v\| = 7.07}\)
Explanation
- Calculating the Sum of Vectors u and v
We are given two vectors, \(u = \langle -4, 7 \rangle\) and \(v = \langle 11, -6 \rangle\). To find the sum \(u + v\), we add the corresponding components of the vectors: the x-components and the y-components separately. Specifically, the sum vector \(w = u + v\) is defined as \(\langle u_x + v_x, u_y + v_y \rangle\). By substituting the given values, we have \(u + v = \langle -4 + 11, 7 + (-6) \rangle\). Evaluating these sums gives \(7\) for the x-component and \(1\) for the y-component, so \(u + v = \langle 7, 1 \rangle\).
- Calculating the Magnitude of the Sum Vector
The magnitude of a vector \(w = \langle w_x, w_y \rangle\), denoted by \(\|w\|\), is calculated using the Pythagorean theorem as \(\|w\| = \sqrt{w_x^2 + w_y^2}\). For our sum vector \(u + v = \langle 7, 1 \rangle\), the magnitude is:
\[\|u + v\| = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50}\]
Calculating the square root of \(50\), we get approximately \(7.071\). Rounding to the nearest hundredth, the magnitude is \(7.07\).
Examples
Vector addition is essential in physics and navigation to determine the net force or resultant displacement. For example, if a ship is traveling due east at a certain velocity and faces a crosswind pushing it north, vector addition allows the captain to calculate the resultant velocity vector and its speed (magnitude), ensuring the vessel stays on the intended course. This is a fundamental application of combining directed quantities in real-world engineering and travel.