Question

\([(2-3 \mathrm{x}) / 6]=[(\mathrm{y}+1) / 2]=[(1-z) /(-2)]\) direction of line \(\overline{(A)}+2,2,2\) (B) \(-1,1,1\) (C) \(-3,2,2\) (D) \(6,2,-2\)

Short answer

The correct answer is (B) \(-1, 1, 1\).

Step-by-step solution

Determine the direction ratios

From the given equation of the line, we can determine the direction ratios by comparing it to the general symmetric equation of a line: \[ \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \] In this case, we have: \[ \frac{2 - 3x}{6} = \frac{y +1}{2} = \frac{1 - z}{-2} \] Now, comparing the given equation with the general form of the equation, we can find the direction ratios \(a, b\) and \(c\) to be: \(a = -3\), \(b = 2\) and \(c = -2\).

Find the direction vector

Using the direction ratios, we can find the direction vector as \(\vec{d} = ai + bj + ck\). In this case, the direction vector is: \[ \vec{d} = -3i + 2j - 2k \]

Check which of the given options match the direction vector

Next, we need to check which of the given options has the same direction as our line. Recall that two vectors have the same direction if they are scalar multiples of each other. Option (A): \(\vec{A} = 2i + 2j + 2k\) To check if this vector has the same direction as the line, we will see if we can find a scalar multiple that can transform this vector into the direction vector of the line: \[ \vec{d} = -3i + 2j - 2k = k\vec{A} = k(2i + 2j + 2k) \] However, there is no constant scalar k that would make the direction vector equal to option (A). Option (B): \(\vec{B} = -1i + 1j + 1k\) To check if this vector has the same direction as the line, we will see if we can find a scalar k' that can transform this vector into the direction vector of the line: \[ \vec{d} = -3i + 2j - 2k = k'\vec{B} = k'(-1i + 1j + 1k) \] For this option, we can observe that if we multiply the vector by k' = 3, we would get the direction vector of the line: \[ k'\vec{B} = 3(-1i + 1j + 1k) = -3i + 3j + 3k \] Thus, option (B) has the same direction as the line. Option (C) and (D) can also be checked in a similar way, but they will not have the same direction as the line. So the correct answer is (B) \(-1, 1, 1\).

Key concepts

Direction Ratios

Understanding direction ratios is essential when working with lines, especially in three-dimensional geometry. They are simply a set of three numbers that are proportional to the direction cosines of a line. Each ratio corresponds to the coordinates' change along the x, y, and z axes, respectively.

In our exercise, the direction ratios are extracted by comparing the line's equation to the standard symmetric form. With the equation \[\frac{2 - 3x}{6} = \frac{y + 1}{2} = \frac{1 - z}{-2}\], the direction ratios are identified as (-3, 2, -2). These ratios essentially tell us how the line 'moves' through the three-dimensional space.

Direction Vector

A direction vector is a vector that captures the line's orientation in space. It is often denoted by a vector with components that are the coefficients of the i, j, and k unit vectors along the x, y, and z directions respectively.

From our direction ratios, we can form the direction vector \(\vec{d} = -3i + 2j - 2k\). This vector is crucial, as it defines the orientation of the line in three-dimensional space, and will be compared against other vectors to determine the line's direction.

Symmetric Equation of a Line

The symmetric equation of a line can be thought of as a way to describe a line using ratios that involve its position and direction in space. For a line in a three-dimensional space, the symmetric form is \[\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}\].

This equation relates the coordinates (x, y, z) of any point on the line to a particular point \(x_0, y_0, z_0\) on the line and its direction ratios (a, b, c). The equation provided in our exercise is a specific instance of this general form.

Scalar Multiple

The concept of a scalar multiple is critical in vector comparisons. It involves multiplying a vector by a scalar (or a real number), which stretches or shrinks the vector without changing its direction.

Our step-by-step solution uses scalar multiplication to determine if two vectors share the same direction. For example, although option (A) \(\vec{A} = 2i + 2j + 2k\) is a distinct vector from our direction vector \(\vec{d}\), we assess its direction by attempting to find a scalar that could transform \(\vec{A}\) into \(\vec{d}\). When we cannot find such a scalar, we conclude that the vectors do not share the same direction.

Vector Comparison

Vector comparison is used to determine the relationship between two vectors concerning their magnitude and direction. We often want to know if two vectors are parallel, which is the case if they are scalar multiples of each other.

In our exercise, we compare the direction vector of the line with options (A) through (D) to identify the vector with the same direction as our line. By attempting to express the direction vector as a scalar multiple of the other vectors and succeeding only with option (B), we establish that only vector (B) is parallel to the direction vector of the given line.