Problem 13 - Calculus: Early Transcendentals

Question

Consider the 12 vectors that have their tails at the center of a clock and their respective heads at each of the 12 digits. What is the sum of these vectors? What if we remove the vector corresponding to 4 o'clock? What if, instead, all vectors have their tails at 12 o'clock, and their heads on the remaining digits?

Step-by-Step Solution

Answer

The sum of all vectors from the center is zero; removing 4 o'clock results in a vector opposite to 4 o'clock; starting from 12, vectors do not cancel out entirely.

Step 1: Understand the Arrangement of Vectors on a Clock Face

Visualize the clock face as a circle with 12 evenly spaced vectors emanating from the center, each pointing toward one of the clock's numbers. Each vector can be thought of as having the same magnitude and a unique direction determined by its position on the clock face (e.g., 1 o'clock, 2 o'clock, etc.).

Step 2: Calculate the Resultant of All Vectors from the Center

Since the vectors are evenly spaced and symmetrical around the center, their directions span the circle completely. Therefore, these vectors form a complete polygon (regular dodecagon) which is symmetric, and the resultant vector becomes zero due to the symmetrical addition of vectors canceling each other out.

Step 3: Modify the Problem by Removing the 4 o'clock Vector

When the vector pointing to 4 o'clock is removed, the symmetry is broken. The resultant vector now points directly opposite to the 4 o'clock vector, as all remaining vectors (11 vectors) are evenly symmetrically distributed around the circle.

Step 4: Analyze the Configuration with Tails at 12 o'clock

If all vectors start from the number 12 and point to each of the other numbers, the vectors are directed as spokes starting from 12, forming segments of the circle. Unlike the first scenario, these vectors do not make a closed symmetrical formation, hence their resultant can be calculated by the vector sum method considering their angular distribution.

Step 5: Evaluate the Sum with Respectivity to the Number of Vectors

Considering the vectors as described in the new setup originating from 12, if examined visually or calculated, it reveals that these vectors do not necessarily cancel out entirely, but that would require further calculation by additive projections on a fixed coordinate system (i.e., they have a non-zero resultant but this requires more advanced computation).

Key Concepts

Clock Vectors

Clock vectors can be imagined as arrows pointing from the center of a clock face to each number on the clock. Each vector has the same length, representing unity, but they all point in different directions. These directions are determined by each vector's position on the clock.

• At 1 o'clock, the vector points just a little past the 0-degree line on a circular plane.
• Each following vector proceeds around the circle at 30-degree increments (since a circle has 360 degrees and a clock has 12 numbers).
• The 12 o'clock position aligns perfectly upward at 0 or 360 degrees.

The core idea is understanding how each vector's direction contributes differently to a resultant vector. Observing these vectors lets you visually perceive balance or imbalance in the vector configuration. Remove a vector from this set, and you affect the symmetry and, potentially, the resultant.

Symmetry

Symmetry in clock vectors refers to the equal spacing of the 12 vectors around a clock face. Imagine slicing a circular pie into 12 equal pieces. Each slice or sector of the pie would be equal, balancing each other out around the central point. This is precisely what symmetry in vectors achieves.
When the vectors are symmetrically distributed, as in the full clock vector formation, they can theoretically form a closed shape (a regular dodecagon). In such a case:

• The symmetrical arrangement ensures that opposite vectors cancel out their effects.
• As a result, the net or resultant vector is zero.
• This perfect balance is disrupted if a single vector (like the one pointing to 4 o'clock) is removed, unbalancing the configuration.

This concept is foundational in vector addition and demonstrates the importance of symmetry in achieving equilibrium.

Vector Resultant

The vector resultant is simply the single vector that represents the sum of all the individual vectors acting at a point. In the context of clock vectors, when all are present and symmetrically arranged, the resultant is zero.

• Each vector in its symmetry counteracts another.
• When we remove one vector, like the one at 4 o'clock, the remaining vectors no longer perfectly counterbalance in all directions.
• The resultant vector of the remaining configuration points in the direction opposite to the removed vector.

To find the resultant in any vector set: - Add vector magnitudes using appropriate rules. - Consider direction, using components if the vectors are not aligned in even directions. This process helps students understand practical vector addition beyond theoretical exercises.

Angular Distribution

Angular distribution describes how vectors are spread out around a central point, like the center of a clock. In the clock face scenario, this distribution is regular and equal at 30-degree increments between each number.
If you imagine each vector originating from the 12 o'clock point instead, the angular distribution shifts. Now, each vector points directly to another number:

• This setup doesn’t allow for perfect symmetry.
• The 30-degree equal spacing is retained, but the symmetry is limited to a semi-circle.
• In this scenario, vectors might add up to a non-zero resultant, unlike the full symmetrical circle from the center.

Calculating such distributions involves understanding how each vector’s angle affects the sum, considering both magnitude and direction. This is crucial when trying to find out how the resultant might shift or change when modifying vector origins or removing vectors.