If \( \alpha \), \( \beta \) and \( \gamma \) are the angles which the vector \( \overrightarrow{OP} \) (O being the origin) makes with positive direction of the coordinate axes, then which of the following are correct?
1. \( \cos^{2}\alpha + \cos^{2}\beta = \sin^{2}\gamma \)
2. \( \sin^{2}\alpha + \sin^{2}\beta = \cos^{2}\gamma \)
3. \( \sin^{2}\alpha + \sin^{2}\beta + \sin^{2}\gamma = 2 \)
Select the correct answer using the code given below.
The angle between the lines \( x + y - 3 = 0 \) and \( x - y + 3 = 0 \) is \( \alpha \) and the acute angle between the lines \( x - \sqrt{3}y + 2\sqrt{3} = 0 \) and \( \sqrt{3}x - y + 1 = 0 \) is \( \beta \). Which one of the following is correct?
Let \( \vec{\alpha} = \hat{i} + 2\hat{j} - \hat{k} \), \( \vec{\beta} = 2\hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{\gamma} = 2\hat{i} + \hat{j} + 6\hat{k} \) be three vectors. If \( \vec{\alpha} \) and \( \vec{\beta} \) are both perpendicular to the vector \( \vec{\delta} \) and \( \vec{\delta} \cdot \vec{\gamma} = 10 \), then what is the magnitude of \( \vec{\delta} \)?
A force \( \vec{F} = \hat{i} + 3\hat{j} + 2\hat{k} \) acts on a particle to displace it from the point \( A(\hat{i} + 2\hat{j} - 3\hat{k}) \) to the point \( B(3\hat{i} - \hat{j} + 5\hat{k}) \). The work done by the force will be
A man running round a racecourse notes that the sum of the distances of two flag-posts from him is always 10 m and the distance between the flag-posts is 8 m. The area of the path he encloses is
