To find a vector that has the same direction as the vector (6, 4, 2) but with a specific length (in this case, 6), you’ll need to follow these steps: Calculate the length (magnitude) of the original vector: The length of a vector (x, y, z) can be calculated using the formula: length = √(x² … Read more
To calculate the length of the arc on a circle that is intercepted by a central angle, you can use the formula: Arc Length (L) = r × θ Where: L = Length of the arc r = Radius of the circle θ = Central angle in radians However, in your case, the central angle … Read more
The function g(x) = x² + 12 is a quadratic function, where the term x² determines its shape. Since the coefficient of x² is positive, the parabola opens upwards. To find the range of this function, we first need to determine its vertex. The vertex of a parabola defined by the function y = ax² … Read more
Calculating the average of all odd numbers up to 100 is a straightforward process! Let’s break it down step by step: 1. **Identify the Odd Numbers:** The odd numbers from 1 to 100 are: 1, 3, 5, 7, …, 99. This can be represented as an arithmetic sequence where: The first term (a) = 1 … Read more
The Law of Sines is a useful formula in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. To find the missing side ‘b’ of a triangle using the Law of Sines, you need some information about the triangle, such as the lengths of one or two … Read more
To find the inverse tangent (or arctan) of a value using the concept of the unit circle, we first need to clarify the context of our question. The arctangent function returns the angle whose tangent is a given number. In this case, you are asking for the angle whose tangent is 13. However, it’s important … Read more
Understanding the Conic Section To identify the conic section represented by the given equation, we need to first rearrange it into a standard form. The equation in question is: x² + 4x + y² + 4y – 4 = 12 We will start by moving all terms to one side of the equation: x² + … Read more
The equation of the parabola is given as x² = 12y. This is a standard form of a vertical parabola. To analyze the properties of this parabola, we can identify key components: 1. Vertex The vertex of a parabola in the form x² = 4py is located at the origin (0, 0). For our equation, … Read more
Understanding the Parabola A parabola is a U-shaped curve that can open upwards, downwards, left, or right. It is defined as the set of points that are equidistant from a point called the focus and a line known as the directrix. Finding the Equation To derive the equation of a parabola from its focus and … Read more