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The image contains a math problem related to vectors and geometry in three-dimensional space. Here’s a breakdown of the parts that I can see clearly:

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**Q1 (a)**:  
A rope starts at the hook and goes in the direction \(\begin{pmatrix} 3 \\ 1 \\ 3 \end{pmatrix}\). It is 8 meters long and ends at point \(P\).

- Find the exact coordinates of point \(P\).

**Q1 (b)**:  
A rope \(D\) starts at the hook and goes in the direction \(\mathbf{i} + \mathbf{j}\). Another rope \(E\) is installed which passes through points \(Q(1, 1, 2)\) and \(R(1, -2, a)\).

(i) Find the values of \(a\) for which \(D\) and \(E\) will **not** meet.

(ii) After considering the artistic effects, the artist uses \(a = -2\). The artist shines a light onto \(E\) perpendicularly onto the wall (which is the \(y-z\) plane). Find the length of the shadow cast by \(QR\) onto the wall.

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Do you want me to start solving part **Q1 (a)** first and proceed step by step?

An artist decides to use light and ropes to create an art installation in a room. Points (x, y, z) are defined relative to a hook of negligible size on the wall at (0, 0, 0), where units are in meters. Ropes are laid in straight lines and the thickness of ropes can be neglected.

Q1 (a): A rope starts at the hook and goes in the direction (3, 1, 3). It is 8 meters long and ends at point P. Find the exact coordinates of P.

Q1 (b): A rope D starts at the hook and goes in the direction (i + j). Another rope E is installed, passing through points Q(1, 1, 2) and R(1, -2, a). (i) Find the values of a for which D and E will not meet. (ii) After considering the artistic effects, the artist uses a = -2. The artist shines a light onto E perpendicularly onto the wall, which is the y-z plane. Find the length of the shadow cast by QR onto the wall.

The artist also wants to add two ropes F and G. Both ropes pass through point Q and are reflections of each other about rope E. It is known that rope F goes in the direction (10, 3, 0). Find the vector equation of G.

This problem involves using vectors in 3D geometry to find the exact coordinates of points, project shadows onto planes, and solve for vector equations. Includes analysis of reflections and vector directions.

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