Final answer:
The correct answer is 'd. none of these', since the projection of the line segment AB onto a line with direction ratios proportional to 6, 2, 3, calculations lead to a dot product of 22, but the answer options provided are incorrect as they do not match this calculation.
Step-by-step explanation:
The question involves finding the projection of the line segment joining points A(-1, 0, 3) and B(2, 5, 1) onto a line with direction ratios proportional to 6, 2, 3. To solve this, we first find the direction vector of AB by subtracting the coordinates of A from B, which gives us AB = (3, 5, -2).
The direction ratios of the line are 6, 2, 3 which also serve as the components of the unit direction vector along the line, let's call it u, but we need to convert them into a unit vector. Assuming u is already a unit vector for simplicity, we then compute the dot product of AB and u, and divide by the magnitude of u (which is 1 in this case because we assumed u to be a unit vector).
The dot product AB • u is given by (3*6 + 5*2 + (-2)*3) which equals 18 + 10 - 6 = 22. Hence, the projection of the line segment AB onto the line is the length of this dot product, which is 22. Since we considered u as a unit vector, its magnitude is 1, so the projection is simply the dot product value itself, which makes 22/7 (considering the division by the magnitude of u) an incorrect choice.
The correct answer should represent the projection length, but without the correct values or options provided, we would deduce it as none of these.