The vectors \vec{a}, \vec{b} and \vec{c} are of the same length. If taken pairwise, they form equal angles. If \vec{a}=\hat{i}+\hat{j} and \vec{b}=\hat{j}+\hat{k}, then what can \vec{c} be equal to? I. \hat{i}+\hat{k} II. \frac{-\hat{i}+4\hat{j}-\hat{k}}{3} Select the correct answer using the code given below.

Explanation

Both \vec{a} and \vec{b} have length \sqrt{2}. The angle between them satisfies \cos\theta = \frac{1}{2}. For option I, \vec{c} = \hat{i}+\hat{k} has length \sqrt{2} and yields \vec{a}\cdot\vec{c}=1 and \vec{b}\cdot\vec{c}=1, so it forms the same angle. For option II, \vec{c} = \frac{-\hat{i}+4\hat{j}-\hat{k}}{3} has length \frac{\sqrt{1+16+1}}{3} = \sqrt{2}, and yields \vec{a}\cdot\vec{c} = \frac{-1+4}{3} = 1 and \vec{b}\cdot\vec{c} = \frac{4-1}{3} = 1. Both vectors satisfy the given conditions.

Related questions on Vector Algebra

• PQRS is a parallelogram. If \vec{PR}=\vec{a} and \vec{QS}=\vec{b}, then what is \vec{PQ} equal to?
• Let \vec{a} and \vec{b} are two unit vectors such that \vec{a}+2\vec{b} and 5\vec{a}-4\vec{b} are <strong>PERPENDICULAR</strong>. Wh...
• Let \vec{a}, \vec{b} and \vec{c} be unit vectors lying on the same <strong>COPLANAR</strong> plane. What is $\{(3\vec{a}+2\vec{b})\tim...
• What are the values of x for which the angle between the vectors 2x^{2}\hat{i}+3x\hat{j}+\hat{k} and \hat{i}-2\hat{j}+x^{2}\hat{k} is ...
• The position vectors of vertices A, B and C of triangle ABC are respectively \hat{j}+\hat{k}, 3\hat{i}+\hat{j}+5\hat{k} and $3\h...