Question

If the direction co-side between two lines are \((3,4,-6)\) and \((9,2,1)\). Then angle between two line is (A) \(\cos ^{-1}[20 /(5246)]\) (B) \(\overline{\cos ^{-1}}[\sqrt{(29) / \sqrt{(5246})]}\) (C) \(\cos ^{-1}[29 / \sqrt{(5246)}]\) (D) \(\cos ^{-1}[\sqrt{(29)} /(5246)]\)

Short answer

Answer: \(\boxed{\text{(C) }\cos^{-1}\left(\frac{29}{\sqrt{5246}}\right)}\)

Step-by-step solution

Find the direction ratios of the two lines

Given direction co-sides are \((3,4,-6)\) and \((9,2,1)\). These are also equal to the direction ratios of the lines. So, the direction ratios of lines are: Line 1: \(a_1=3\), \(b_1=4\), and \(c_1=-6\). Line 2: \(a_2=9\), \(b_2=2\), and \(c_2=1\).

Use the formula to find the angle between the lines

The formula for the angle between two lines with direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) is given by: \(\cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{(a_1^2 + b_1^2 + c_1^2)(a_2^2 + b_2^2 + c_2^2)}}\) Now we will substitute the values and find \(\cos \theta\): \(\cos \theta = \frac{3(9) + 4(2) + (-6)(1)}{\sqrt{((3^2 + 4^2 + (-6)^2)(9^2 + 2^2 + 1^2))}}\)

Calculate the values

Simplify the numerator and denominator separately: Numerator: \(27 + 8 - 6 = 29\) Denominator: \(\sqrt{(9 + 16 + 36)(81 + 4 + 1)} = \sqrt{(61)(86)} = \sqrt{5246}\) Now, compute \(\cos \theta\): \(\cos \theta = \frac{29}{\sqrt{5246}}\)

Find the angle and match the answer

To find the angle between the two lines, we need to calculate the inverse cosine of the value we obtained in the previous step: \(\theta = \cos^{-1}\left(\frac{29}{\sqrt{5246}}\right)\) This matches with option (C) in the list of possible answers. Answer: \(\boxed{\text{(C) }\cos^{-1}\left(\frac{29}{\sqrt{5246}}\right)}\)

Key concepts

Direction Ratios

Direction ratios are fundamental in understanding the orientation of lines in three-dimensional space. They are essentially the components that define the direction of a line. Given a line in a 3D space, you describe it using a set of direction ratios: \(a_1, b_1, c_1\) for the first line and \(a_2, b_2, c_2\) for the second line. These can be thought of as multiples that give proportionate directions along the x, y, and z axes.

For the lines given in our problem, we had direction co-sides \(3, 4, -6\) and \(9, 2, 1\). These co-sides directly translate into direction ratios for the respective lines:

• Line 1: \(a_1 = 3\), \(b_1 = 4\), \(c_1 = -6\)
• Line 2: \(a_2 = 9\), \(b_2 = 2\), \(c_2 = 1\)

These ratios help us identify how the line traverses through space by multiplication with a scalar to extend or reduce its length while maintaining direction.

Cosine Formula

The cosine formula for finding the angle between two lines in 3D is derived from vector dot products. This is an essential concept when working with angles in mathematics and physics. Given two lines with direction ratios, the angle \(\theta\) between them can be found using the formula:

• \[\cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{(a_1^2 + b_1^2 + c_1^2)(a_2^2 + b_2^2 + c_2^2)}}\]

In this equation, the numerator is the dot product of the direction ratios, which reflects how much the two directions "overlap." The denominator involves the magnitudes (or lengths) of the direction vectors, which normalizes the dot product, giving us a dimensionless number between -1 and 1, known as the cosine of the angle.

Substituting the specific values from the problem:

• Numerator: \(29\)
• Denominator: \(\sqrt{5246}\)

Therefore, \(\cos \theta = \frac{29}{\sqrt{5246}}\). This form tells us how close the two lines are to being parallel (cosine near 1) or perpendicular (cosine near 0).

Inverse Cosine Function

Once you've determined the cosine of the angle between two lines, you need to find the actual angle, often done using the inverse cosine function, written as \(\cos^{-1}\) or "arccos." This function transforms a cosine value back to an angle in radians or degrees.

The inverse cosine function is key when interpreting precise angles since \(\cos^{-1}(x)\) essentially finds the unique angle \(\theta\) that lies between 0 and \(\pi\) (180 degrees) for which \(\cos(\theta) = x\). This makes it extremely useful in trigonometry and geometry.

In our problem:

• \[\theta = \cos^{-1}\left(\frac{29}{\sqrt{5246}}\right)\]

This calculation provides us the specific angle between the two lines, which in this case matched option (C). Understanding how to move from a cosine value to an angle with \(\cos^{-1}\) is critical in accurately interpreting spatial relationships.