Given that the vectors $u$, $v$, and $w$ satisfy $\left|u\right|=\left|v\right|=1$ and $u+v+w=0$, what is the magnitude of $w$?

• A. $\sqrt{3}$
• B. $2$
• C. $1$
• D. $0$

Answer: (A)

Verified step by step guidance

1. Start with the given vector equation: \(\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}\). Rearrange it to express \(\mathbf{w}\) in terms of \(\mathbf{u}\) and \(\mathbf{v}\): \(\mathbf{w} = - (\mathbf{u} + \mathbf{v})\).
2. Recall that the magnitude of a vector and its negative are the same, so \(|\mathbf{w}| = |\mathbf{u} + \mathbf{v}|\).
3. Use the formula for the magnitude of the sum of two vectors: \(|\mathbf{u} + \mathbf{v}| = \sqrt{|\mathbf{u}|^2 + |\mathbf{v}|^2 + 2 |\mathbf{u}| |\mathbf{v}| \cos \theta}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\).
4. Substitute the given magnitudes \(|\mathbf{u}| = 1\) and \(|\mathbf{v}| = 1\) into the formula: \(|\mathbf{w}| = \sqrt{1^2 + 1^2 + 2 \cdot 1 \cdot 1 \cdot \cos \theta} = \sqrt{2 + 2 \cos \theta}\).
5. To find \(|\mathbf{w}|\), determine the value of \(\cos \theta\) using the condition that \(\mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0}\), which implies \(\mathbf{w} = - (\mathbf{u} + \mathbf{v})\). This means \(\mathbf{w}\) is opposite to \(\mathbf{u} + \mathbf{v}\), so the vectors form a closed triangle. Use this geometric insight to find \(\cos \theta\) and then compute \(|\mathbf{w}|\).