Problem 18 - Applied Mathematics

Question

Two vectors of equal magnitude and which are in the same direction are. equal vectors.

Step-by-Step Solution

Answer

Equal vectors.

Step 1: Understand Vector Properties

Two vectors are equal if they have both the same magnitude and the same direction. Review vector properties to verify.

Step 2: Analyze Given Information

The exercise states that there are two vectors with equal magnitudes and they are in the same direction.

Step 3: Apply Vector Equality Condition

According to vector properties, vectors that have the same magnitude and are in the same direction are considered equal.

Step 4: Conclude the Solution

Since the given vectors meet the conditions of having equal magnitudes and directions, they are indeed equal vectors.

Key Concepts

Vector Properties

Vectors are fundamental elements in physics and mathematics. They are often represented as arrows with both length and direction. There are several properties of vectors that one should be aware of:

• Magnitude: This is the length of the vector. It is a measure of how large or small the vector is.
• Direction: This is the orientation of the vector in space. It shows where the vector is pointing.
• Equality: Two vectors are considered equal if they have the same magnitude and direction.
• Scalar Multiplication: Multiplying a vector by a scalar (a real number) will change its magnitude but not its direction, unless the scalar is negative, in which case it also reverses the direction.

Understanding these properties helps in visualizing and manipulating vectors in various applications. For instance, if two vectors share both a common direction and identical magnitude, we can confidently say they are equal vectors.

Vector Magnitude

The magnitude of a vector is essentially its length. It's often denoted as \| \mathbf{v} \| for a vector \mathbf{v}\. To calculate the magnitude of a vector \mathbf{v} = (x, y)\ in a 2-dimensional space, you can use the formula:

\[ \| \mathbf{v} \| = \sqrt{x^2 + y^2} \] For a vector \mathbf{v} = (x, y, z)\ in a 3-dimensional space, the formula becomes: \[ \| \mathbf{v} \| = \sqrt{x^2 + y^2 + z^2} \] Magnitude is a crucial property because it tells you the 'size' of the vector. Whenever you are comparing vectors to determine if they are equal, you first need to check if their magnitudes are identical. Only then can you move on to compare their directions.

Vector Direction

The direction of a vector indicates where the vector is pointing. It can often be described by the angle it makes with a reference axis, usually the positive x-axis. To find the direction of a 2-dimensional vector \mathbf{v} = (x, y)\, you can use the formula:

\[ \theta = \tan^{-1} \left(\frac{y}{x}\right) \] Here, \theta \ is the angle the vector makes with the positive x-axis. In three-dimensional space, describing direction becomes a bit more complex, often requiring both azimuth and elevation angles or using directional cosines. To confirm the equality of two vectors, it's essential to ensure that not only their magnitudes match but also the vector directions align perfectly. Only then can the vectors be considered truly equal.