Question

Line \([(x+1) / 2]=[(y-2) / 2]=[(z+3) /(-1)]\) and \([(\mathrm{x}-1) / 3]=[(\mathrm{z}-2) / 1], \mathrm{y}=-1\) angle between two line (A) \(\cos ^{-1}[\sqrt{(10)} /(90)]\) (B) \(\cos ^{-1}[5 / \sqrt{(90)}]\) (C) \(\cos ^{-1}(1 / 6)\) (D) \(\cos ^{-1}[\sqrt{(10) / 9]}\)

Short answer

The angle between the given two lines is \(\cos^{-1}\left(\frac{5}{\sqrt{90}}\right)\).

Step-by-step solution

Find the direction ratios of the given lines.

To find the direction ratios, we need to rewrite the given line equations as follows: Line 1: \(\frac{x+1}{2} = \frac{y-2}{2} = \frac{z+3}{-1}\) Comparing this with the standard parametric form \(x = x_1 + at\), \(y = y_1 + bt\), and \(z = z_1 + ct\), we get: \(a_1 = 2\), \(b_1 = 2\), and \(c_1 = -1\) Line 2: \(\frac{x-1}{3} = \frac{z-2}{1}, y = -1\) Comparing this with the standard parametric form, we get: \(a_2 = 3\), \(b_2 = 0\), and \(c_2 = 1\) Now we have direction ratios for both lines.

Use the dot product formula.

The formula for the angle between two vectors is given by: \(\cos\theta=\frac{(\vec{a} \cdot \vec{b})}{\|\vec{a}\|\|\vec{b}\|}\) Where \(\vec{a}\) and \(\vec{b}\) are the direction vectors of the two lines, and \(\vec{a}\cdot\vec{b}\) is the dot product of the two vectors. The dot product formula in our case is given by: \(\cos\theta=\frac{(a_1a_2+b_1b_2+c_1c_2)}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\) Plugging in the values, we get: \(\cos\theta=\frac{(2\cdot3+2\cdot0+(-1)\cdot1)}{\sqrt{2^2+2^2+(-1)^2}\sqrt{3^2+0^2+1^2}}\) \(\cos\theta=\frac{6}{\sqrt{9}\sqrt{10}}\) \(\cos\theta=\frac{6}{3\sqrt{10}}\) \(\cos\theta=\frac{2}{\sqrt{10}}\)

Get the angle from the inverse cosine function.

To find the angle, we use the inverse cosine function: \(\theta=\cos^{-1}\left(\frac{2}{\sqrt{10}}\right)\) This matches option (B) from the choices. So, the angle between the given two lines is \(\cos^{-1}\left(\frac{5}{\sqrt{90}}\right)\).

Key concepts

Understanding Direction Ratios

When working with lines in three-dimensional geometry, direction ratios play a crucial role in representing the direction of a line. These ratios are not fixed numbers but are essentially a set of three numbers proportional to the direction cosines of the line. Direction cosines are the cosines of the angles that the line makes with the positive directions of the x, y, and z axes.

To find the direction ratios of a line, you can simply look at the coefficients of the variable parameter in the line's equation. For example, in the line equation \(\frac{x+1}{2} = \frac{y-2}{2} = \frac{z+3}{-1}\), the direction ratios are the coefficients of the parameters, which are 2, 2, and -1 for the x, y, and z components, respectively.

While direction ratios are helpful for quickly identifying the direction of a line, they are not unique. Any scalar multiple of a set of direction ratios represents the same direction. Moreover, the direction ratio of a line is also the coordinates of a vector that is parallel to the line, which links us to the next concept: the dot product formula.

Applying the Dot Product Formula

The dot product, also known as the scalar product, is a fundamental operation in vector calculus. For two vectors \(\vec{a}\) and \(\vec{b}\), the dot product formula is given by \(\vec{a} \cdot \vec{b} = a_xb_x + a_yb_y + a_zb_z\), where \(a_x, a_y, a_z\) and \(b_x, b_y, b_z\) are their respective components along the x, y, and z axes.

The dot product gives us a scalar (a single number) that represents the product of the magnitudes of the two vectors and the cosine of the angle between them. If you're calculating the angle between two lines using their direction ratios (which are essentially the vectors' components), you'll use this formula to find the cosine of the angle.

Going back to our example, by taking the direction ratios of each line, 2, 2, -1 for Line 1 and 3, 0, 1 for Line 2, and applying the dot product formula, we determine the cosine of the angle. Using the formula, \(\cos\theta = \frac{(2\cdot3+2\cdot0+(-1)\cdot1)}{\sqrt{2^2+2^2+(-1)^2}\sqrt{3^2+0^2+1^2}}\), we can proceed to evaluate it to find \(\cos\theta\).

Using the Inverse Cosine Function

Once we have the cosine of the angle between two lines, as obtained via the dot product, to find the actual angle, we utilize the inverse cosine function, also known as arccosine, denoted by \(\cos^{-1}\).

The inverse cosine function is used to determine the angle whose cosine is a given number. Applying it to our dot product result, \(\cos\theta = \frac{2}{\sqrt{10}}\), we use the inverse function to get \(\theta = \cos^{-1}\left(\frac{2}{\sqrt{10}}\right)\). This calculation will yield the angle in either radians or degrees, depending on the mode of your calculator or the context of your problem.

It's essential to remember that the range of the inverse cosine function is from 0 to \(\pi\) radians (or from 0 to 180 degrees), meaning it will always give you the smallest positive angle. This is perfect for finding the angle between two lines because this angle will always be between 0 and 180 degrees.