Question

\(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) are two vectors \(\mathrm{U}_{\mathrm{A}}^{\wedge}=\mathrm{U}_{\mathrm{B}}\) Now find the true option. (A) \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) equal vectors (B) \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) are in opposite direction (C) \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) are in same direction (D) \(\mathrm{A}^{\rightarrow} \perp \mathrm{B}^{\rightarrow}\)

Short answer

The correct answer is option (C): \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) are in the same direction.

Step-by-step solution

A unit vector is a vector that has the same direction as the original vector, but a magnitude of 1. So, if two vectors have the same unit vector, it means they have the same direction and the only difference is in their magnitudes. #Step 2: Analize the given information#

Given that the unit vectors of vectors \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) are equal (\(\mathrm{U}_{\mathrm{A}}^{\wedge}=\mathrm{U}_{\mathrm{B}}\)), it means that their directions are the same. Now, we'll check the given options to find the true answer: #Step 3: Analyze option (A)#

(A) says that both vectors are equal. However, the only information we have is that their directions are the same, so we don't have enough evidence to claim they are equal. Hence, option (A) is not the true answer. #Step 4: Analyze option (B)#

(B) suggests that both vectors are in the opposite direction. However, since their unit vectors are equal, it means they have the same direction, not the opposite direction. Thus, option (B) is not the true answer. #Step 5: Analyze option (C)#

(C) states both vectors are in the same direction. This is consistent with the given information that their unit vectors are equal. Therefore, option (C) is the true answer. #Step 6: Analyze option (D) just for completeness#

(D) proposes that the vectors are perpendicular. However, since their unit vectors are equal, they must have the same direction, not being perpendicular. Therefore, option (D) is not the true answer. In conclusion, the correct answer to the exercise is option (C): \(\mathrm{A}^{\rightarrow}\) and \(\mathrm{B}^{\rightarrow}\) are in the same direction.

Key concepts

Unit Vector

A unit vector is a special type of vector that points in the same direction as the original vector but has a magnitude (length) of exactly one unit. This concept makes it very useful for understanding the direction of a vector independent of its magnitude. Imagine breaking down a vector into parts; the unit vector gives you just the "direction piece."
Unit vectors maintain the direction of the original, but because they have a magnitude of 1, they don’t give any information about the "size" or "length" of the original vector. For example, if you have a vector pointing to the north, its unit vector will also point to the north, but it will only stretch as far as 1 unit.
The utility of unit vectors is tremendous in mathematics and physics, where they are often used to simplify calculations, as they isolate directional components. The notation commonly used for a unit vector scaled from a vector \( \mathbf{A} \) is \( \hat{\mathbf{A}} \), with the "hat" symbol \( \hat{} \) denoting that it's a unit vector.

Vector Direction

Vector direction is a fundamental concept in understanding how vectors work because it gives you the "point" a vector aims towards in a space. Direction doesn’t concern itself with how long the vector is but rather with which way it points. If you think of vectors as arrows, then direction is simply where the arrowhead points.
When multiple vectors have the same direction, they can either be equal, if their magnitudes are identical, or they can vary in length, indicating different magnitudes. For vectors to be described as having the same direction, their unit vectors must be identical because a unit vector only deals with direction by ensuring uniformity in magnitude.
Understanding vector direction is crucial when analyzing forces, movements, and even lines in geometry, as direction is often the first property used to categorize and solve problems. Don’t forget: while vectors can originate at different points, it’s their direction that aligns them!

Vector Magnitude

The magnitude of a vector is essentially its "length" or "size." It's a scalar quantity, meaning it only has magnitude and no direction. When you see a vector drawn as an arrow, the magnitude is how long the arrow is. The larger the magnitude, the more forceful or pronounced that vector’s properties are.
To find a vector's magnitude mathematically, you take the square root of the sum of the squares of its components. If you have a vector \( \mathbf{A} = (x, y, z) \), its magnitude \( |\mathbf{A}| \) is calculated as:\[|\mathbf{A}| = \sqrt{x^2 + y^2 + z^2}\]
Understanding magnitude is key to interpreting what a vector is representing. For example, in physics, the magnitude of a displacement vector tells you how far you've traveled, while in force vectors, the magnitude tells you how strong the force is. When analyzing vectors, it’s important to simultaneously consider both their magnitude and direction for a complete understanding.