(b) Opposite sides of a square are along the lines:
r=i^+2j^−4k^+λ(2i^+3j^+6k^)
r=3i^+3j^ −5 k^+μ(2i^+3j^+6k^) Find the area of the square if direction ratios of the other pair of opposite sides of the square are given by <-3, 6, p>. Also, find the value of p.
(square, vector equation, direction ratios, area, parallel lines)
Get the complete, step-by-step math solution for: "(b) Opposite sides of a square are along the lines: {r} = {i} + 2 {j} - 4 {k} + (2 {i} + 3 {j} + 6 {k}) {r} = 3 {i} + 3 {j} - 5 {k} + (2 {i} + 3 {j} +...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify Parallel Lines and Direction Vectors
The given equations are for two lines. Since they are opposite sides of a square, they must be parallel. This means their direction vectors, b1 and b2, must be parallel. We can observe this from the given equations.
Step 2: Extract Position Vectors and Direction Vectors
From the given line equations of the form r=a+tb, we identify the position vector of a point on each line (a1 and a2) and their common direction vector (b). These lines are L1 and L2 respectively.
Step 3: Calculate Perpendicular Distance Between Parallel Lines
The side length of the square, let's call it s, is equal to the perpendicular distance d between the two parallel lines. We use the formula for the distance between two parallel lines. Here, (a2−a1) is a vector connecting a point on L1 to a point on L2, and b is the common direction vector.
Step 4: Compute (a2−a1)
First, we find the difference between the position vectors a2 and a1. This vector goes from a point on the first line to a point on the second line.
Step 5: Compute the Cross Product (a2−a1)×b
Next, we calculate the cross product of the vector (a2−a1) and the direction vector b. The magnitude of this cross product is used in the distance formula.
Step 6: Compute Magnitude of Cross Product
We find the magnitude of the resulting vector from the cross product. This magnitude forms the numerator of our distance formula.
Step 7: Compute Magnitude of Direction Vector b
We calculate the magnitude of the common direction vector b. This magnitude forms the denominator of our distance formula.
Step 8: Calculate the Side Length of the Square (s)
Now we substitute the calculated magnitudes into the distance formula. This gives us the perpendicular distance between the two lines, which is the side length (s) of the square.
Step 9: Calculate the Area of the Square
The area of a square is given by the square of its side length. We square the calculated side length s to find the area.
Step 10: Determine the Direction Ratios of Perpendicular Lines
The other pair of opposite sides of the square must be perpendicular to the given pair of lines. Therefore, their direction vector, which is given by <−3,6,p>, must be perpendicular to the direction vector of the first pair of lines, b=<2,3,6>. The dot product of two perpendicular vectors is zero.
Step 11: Solve for p
We set up the dot product equation and solve for p. This algebraic manipulation gives us the numerical value of p which completes the direction ratios of the other pair of sides.