Question

find the number of vectors in \(\mathrm{R}^{3}\) such the angle between X-axis and vectors are \((\pi / 3)\) (a) 1 (b) 2 (c) 4 (d) infinite times

Short answer

\( \text{Answer: } (d) \text{ infinite times}\)

Step-by-step solution

Understand the Dot Product Formula

To find the angle between two vectors, we use the dot product formula, which is: \[ \boldsymbol{A} \cdot \boldsymbol{B} = |\boldsymbol{A}| \cdot |\boldsymbol{B}| \cdot \cos\theta \] where A and B are two vectors, ¦A¦ and ¦B¦ are their magnitudes, θ is the angle between them, and A · B is the dot product of the vectors.

Define the vectors and angle

Let's consider a vector in ℝ³ as \(\boldsymbol{V} = (x, y, z)\) and the X-axis represented as a vector \(\boldsymbol{X}\) with components (1,0,0). The given angle between the X-axis and the vector is \(\frac{\pi}{3}\) radians. So, we can write the dot product formula for these vectors as: \[ \boldsymbol{V} \cdot \boldsymbol{X} = |\boldsymbol{V}| \cdot |\boldsymbol{X}| \cdot \cos\frac{\pi}{3} \]

Calculate the dot product and magnitudes

The dot product of vector V and vector X is given by: \[ \boldsymbol{V} \cdot \boldsymbol{X} = (x, y, z) \cdot (1, 0, 0) = x \] The magnitude of vector V is: \[ |\boldsymbol{V}| = \sqrt{x^2 + y^2 + z^2} \] The magnitude of the X-axis vector X is: \[ |\boldsymbol{X}| = \sqrt{1^2 + 0^2 + 0^2} = 1 \]

Substitute values in the dot product formula

Now, substitute the values of magnitudes and dot product in the dot product formula: \[ x = \sqrt{x^2 + y^2 + z^2} \cdot 1 \cdot \cos\frac{\pi}{3} \] We know that \(\cos\frac{\pi}{3} = \frac{1}{2}\), so the equation becomes: \[ x = \frac{1}{2} \sqrt{x^2 + y^2 + z^2} \]

Solve the equation

Square both sides of the equation: \[ x^2 = \frac{1}{4}(x^2 + y^2 + z^2) \] Then, \[ 3x^2 = y^2 + z^2 \] As per this equation, for any given x, we can find y and z such that y² + z² = 3x², and hence the vector (x,y,z) makes an angle of π/3 with the X-axis.

Determine the number of vectors

Since there are an infinite number of (x, y, z) triplets that satisfy the equation \(3x^2 = y^2 + z^2\), there are an infinite number of vectors in ℝ³ that make an angle of π/3 with the X-axis. So, the correct answer is (d) infinite times.

Key concepts

Dot Product Formula

The dot product formula is essential for calculating the angle between two vectors. It's given by the equation \( \boldsymbol{A} \cdot \boldsymbol{B} = |\boldsymbol{A}| \cdot |\boldsymbol{B}| \cdot \cos\theta \), where \( \boldsymbol{A} \cdot \boldsymbol{B} \) is the dot product of vectors A and B, \( |\boldsymbol{A}| \) and \( |\boldsymbol{B}| \) are the magnitudes of these vectors, and \( \theta \) is the angle between them.

For example, if we want to find the angle between the X-axis and a vector in three-dimensional space, we consider the X-axis as a vector, usually written as \( (1, 0, 0) \) for simplicity. If our vector is \( (x, y, z) \) and we know the angle we're looking for, we can rearrange the dot product formula to solve for whatever variable we need, such as the vector's components or the angle itself. In the context of an exercise, we use this formula as a way to establish a relationship between known and unknown quantities and solve for the specifics.

Vector Magnitudes

Vector magnitudes are a measure of a vector's length or size. For any vector \( \boldsymbol{V} = (x, y, z) \), the magnitude is calculated using the square root of the sum of its components squared: \( |\boldsymbol{V}| = \sqrt{x^2 + y^2 + z^2} \).

This value is crucial when we deal with vectors in various applications like physics, engineering, and mathematics. When calculating the angle between two vectors using the dot product, you must know the magnitudes of the vectors involved. Understanding how to compute vector magnitudes allows you to apply the dot product formula correctly and understand the geometric implications of vector operations. The magnitude reflects the vector's contribution to the dot product, affecting the resulting angle.

Cosine of Angle

The cosine of an angle in the context of vectors is used to determine the directionality between two vectors. Specifically, for the angle \( \theta \) between two vectors, \( \cos\theta \) quantifies the extent to which the vectors point in the same direction.

For an angle \( \theta = \frac{\pi}{3} \) radians, the cosine value is \( \cos\frac{\pi}{3} = \frac{1}{2} \). When this value is used in conjunction with the dot product formula, it allows us to isolate a vector's individual components, as seen in this exercise. The value of the cosine is instrumental in estimating the angle and plays a key role in vector analysis and in solving problems like the one presented in the exercise. Knowing how to use the cosine function gives you a powerful tool to analyze the orientation and relative positioning of vectors in space.