A line makes angles alpha, beta, gamma and delta with the four diagonals of a cube, prove that cos2(alpha)+cos2(beta)+cos2(gamma)+cos2(delta)=4/3.
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Step-by-Step Solution
Step 1: Define Cube Vertices and Diagonals
To solve this problem, we first establish a coordinate system for the cube. Let the side length of the cube be a. We place one vertex at the origin (0,0,0) and align its edges with the coordinate axes. This defines the coordinates of all eight vertices of the cube.
Step 2: Determine Vector Representations of Cube Diagonals
A cube has four main diagonals connecting opposite vertices. We express these diagonals as vectors using the coordinates defined in the previous step. For example, the diagonal connecting (0,0,0) to (a,a,a) is represented by the vector (a,a,a). The other diagonals are found similarly.
Step 3: Define a General Line Vector
Let an arbitrary line make angles α,β,γ,δ with the four diagonals. We can represent this line by a unit vector L=(l,m,n), where l, m, n are its direction cosines. Thus, we have l2+m2+n2=1.
Step 4: Apply Dot Product Formula for Angles
The cosine of the angle between two vectors can be found using their dot product. For a vector A and a vector B, the formula is cosθ=∣A∣∣B∣A⋅B. We will apply this to find the cosines of the angles between L and each of the diagonal vectors.
Step 5: Calculate Cosine Squared for Each Angle
We calculate cos2α, cos2β, cos2γ, and cos2δ using the dot product formula. The magnitude of each diagonal vector is a2+a2+a2=3a2=a3. Since L is a unit vector, ∣L∣2=l2+m2+n2=1. This simplifies the denominators considerably.
Step 6: Sum the Cosine Squared Terms
Now we sum the expressions for cos2α, cos2β, cos2γ, and cos2δ. Each term has a common denominator of 3, so we can combine the numerators.
Step 7: Expand and Simplify Numerators
We expand each squared term (A+B+C)2=A2+B2+C2+2AB+2BC+2CA. This will help us identify terms that cancel out when summed.
Step 8: Perform the Summation
Adding the expanded terms, we observe that the cross-product terms (2lm, 2mn, 2nl) cancel each other out in pairs. For example, 2lm−2lm+2lm−2lm=0. We are left with four times the sum of the squares of the direction cosines.
Step 9: Substitute l2+m2+n2=1
Since l, m, n are the direction cosines of a unit vector along the line, we know that l2+m2+n2=1. Substituting this value simplifies the sum of the numerators to 4.
Step 10: Final Result
Finally, we substitute the simplified numerator back into the expression for the sum of cosine squared terms. This proves the identity that the sum is equal to 4/3.