This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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5. (a) The value of the constant a Step 1: Write the equations of the lines in component form. Given the equations of the lines: r_1 = i + aj - k + (i + 2j + 2k) r_1 = (1+)i + (a+2)j + (-1+2)k r_2 = ai + 2k + (-j + 3k) r_2 = ai - + (2+3)k Step 2: Equate the components since the lines intersect. Equating the i, j, and k components: 1. 1+ = a 2. a+2 = - 3. -1+2 = 2+3 Step 3: Solve the system of equations for a, , and . Substitute equation (1) into equation (2): (1+) + 2 = - 1+3 = - (Equation 4) From equation (4), we can express in terms of : = -1-3 Substitute this expression for into equation (3): -1+2 = 2+3(-1-3) -1+2 = 2-3-9 -1+2 = -1-9 Add 9 to both sides: 11 = 0 = 0 Now substitute =0 back into equation (1) to find a: a = 1+ a = 1+0 a = 1 The value of the constant a is 1. 5. (b) The position vector of the point of intersection Step 1: Substitute the value of (or ) into one of the line equations. Using =0 and a=1 in the equation for r_1: r_1 = (1+0)i + (1+2(0))j + (-1+2(0))k r_1 = 1i + 1j - 1k The position vector of the point of intersection is i + j - k. 5. (c) The cosine of the acute angle between r_1 and r_2 Step 1: Identify the direction vectors of the lines. The direction vector for r_1 is d_1 = i + 2j + 2k. The direction vector for r_2 is d_2 = -j + 3k. Step 2: Calculate the dot product of the direction vectors. d_1 · d_2 = (1)(0) + (2)(-1) + (2)(3) d_1 · d_2 = 0 - 2 + 6 d_1 · d_2 = 4 Step 3: Calculate the magnitudes of the direction vectors. |d_1| = sqrt(1^2 + 2^2 + 2^2) = sqrt(1+4+4) = sqrt(9) = 3 |d_2| = sqrt(0^2 + (-1)^2 + 3^2) = sqrt(0+1+9) = sqrt(10) Step 4: Use the dot product formula to find the cosine of the angle. The cosine of the angle between two vectors is given by = d_1 · d_2|d_1||d_2|. For the acute angle, we take the absolute value of the dot product. = |d_1 · d_2||d_1||d_2| = (|4|)/((3)(sqrt(10))) = (4)/(3sqrt(10)) Rationalize the denominator: = 4sqrt(10)3sqrt(10)sqrt(10) = 4sqrt(10)30 = 2sqrt(10)15 The cosine of the acute angle between r_1 and r_2 is 2sqrt(10)15. What's next?