Analyzing Vector Statements in a Triangle

Let's analyze each given statement about the vectors in a triangle ABC, taken in order. Remember that for a triangle ABC, the sides can be represented by vectors \(\overrightarrow{AB}\), \(\overrightarrow{BC}\), and \(\overrightarrow{CA}\) (or their negatives).

A fundamental principle for vectors in a triangle is the Triangle Law of Vector Addition. This law states that if two sides of a triangle represent two vectors in magnitude and direction taken in order, then the third side of the triangle represents the resultant vector in magnitude and direction taken in the opposite order.

For triangle ABC, this can be expressed as:

• \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\)
• \(\overrightarrow{BC} + \overrightarrow{CA} = \overrightarrow{BA}\)
• \(\overrightarrow{CA} + \overrightarrow{AB} = \overrightarrow{CB}\)

Also, remember that \(\overrightarrow{XY} = -\overrightarrow{YX}\). This means \(\overrightarrow{CA} = -\overrightarrow{AC}\), \(\overrightarrow{AB} = -\overrightarrow{BA}\), and \(\overrightarrow{BC} = -\overrightarrow{CB}\).

We will now evaluate each statement for its correctness in a triangle ABC.

Analyzing Each Vector Statement

Statement 1: \(\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CA} = \vec 0\)

Using the triangle law, we know that \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\). Substituting this into statement 1:

\(\overrightarrow{AC} + \overrightarrow{CA} = \vec 0\)

Since \(\overrightarrow{CA} = -\overrightarrow{AC}\), we replace \(\overrightarrow{CA}\):

\(\overrightarrow{AC} + (-\overrightarrow{AC}) = \vec 0\)

\(\overrightarrow{AC} - \overrightarrow{AC} = \vec 0\)

\(\vec 0 = \vec 0\)

This equation is always true for any triangle. Thus, statement 1 is correct.

Statement 2: \(\overrightarrow {AB} + \overrightarrow {BC} - \overrightarrow {CA} = \vec 0\)

Again, using \(\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}\), substitute into statement 2:

\(\overrightarrow{AC} - \overrightarrow{CA} = \vec 0\)

Substitute \(\overrightarrow{CA} = -\overrightarrow{AC}\):

\(\overrightarrow{AC} - (-\overrightarrow{AC}) = \vec 0\)

\(\overrightarrow{AC} + \overrightarrow{AC} = \vec 0\)

\(2\overrightarrow{AC} = \vec 0\)

This implies \(\overrightarrow{AC} = \vec 0\). For \(\overrightarrow{AC}\) to be the zero vector, points A and C must coincide. If A and C coincide, it does not form a triangle ABC with distinct vertices. Therefore, statement 2 is incorrect for a triangle.

Statement 3: \(\overrightarrow {AB} - \overrightarrow {BC} + \overrightarrow {CA} = \vec 0\)

We can rewrite the statement as \(\overrightarrow {AB} + \overrightarrow {CA} = \overrightarrow {BC}\). From the triangle law, we know that \(\overrightarrow{CA} + \overrightarrow{AB} = \overrightarrow{CB}\). Thus, the statement is equivalent to \(\overrightarrow {CB} = \overrightarrow {BC}\).

Since \(\overrightarrow{CB} = -\overrightarrow{BC}\), the equation becomes:

\(-\overrightarrow{BC} = \overrightarrow{BC}\)

\(2\overrightarrow{BC} = \vec 0\)

This implies \(\overrightarrow{BC} = \vec 0\). For \(\overrightarrow{BC}\) to be the zero vector, points B and C must coincide. If B and C coincide, it does not form a triangle ABC with distinct vertices. Therefore, statement 3 is incorrect for a triangle.

Statement 4: \(\overrightarrow {BA} - \overrightarrow {BC} + \overrightarrow {CA} = \vec 0\)

We can rewrite the statement as \(\overrightarrow {BA} + \overrightarrow {CA} = \overrightarrow {BC}\). From the triangle law, we know that \(\overrightarrow{BA} + \overrightarrow {AC} = \overrightarrow {BC}\). Thus, the statement is equivalent to \(\overrightarrow {BA} + \overrightarrow {CA} = \overrightarrow {BA} + \overrightarrow {AC}\).

This simplifies to \(\overrightarrow {CA} = \overrightarrow {AC}\).

Since \(\overrightarrow{CA} = -\overrightarrow{AC}\), the equation becomes:

\(-\overrightarrow{AC} = \overrightarrow{AC}\)

\(2\overrightarrow{AC} = \vec 0\)

This implies \(\overrightarrow{AC} = \vec 0\). For \(\overrightarrow{AC}\) to be the zero vector, points A and C must coincide. If A and C coincide, it does not form a triangle ABC with distinct vertices. Therefore, statement 4 is incorrect for a triangle.

Summary of Statement Correctness

Let's summarize the analysis in a table:

Conclusion on Correct Statements

Based on the analysis, only statement 1 is correct for a triangle ABC. Therefore, the number of correct statements is one.

Revision Table: Key Triangle Vector Properties

Additional Information: Vector Displacement and Geometry

Vectors are often used to represent displacement. When considering a triangle ABC, the vector \(\overrightarrow{AB}\) represents the displacement from point A to point B, \(\overrightarrow{BC}\) from B to C, and \(\overrightarrow{CA}\) from C to A. If you start at A, move to B, then to C, and finally back to A, your total displacement from the starting point (A) is zero. This is why the sum \(\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA}\) results in the zero vector \(\vec 0\).

This principle is very powerful and applies to any closed path formed by vectors. For instance, if you walk around the perimeter of a park, starting and ending at the same spot, your total displacement is zero, regardless of how far you walked. The vector sum of all the segments of your path would be the zero vector.

Understanding how to manipulate vector equations using properties like the triangle law and the negative vector is fundamental to solving problems in vector algebra and its applications in physics and engineering. Incorrect statements often arise from incorrect application of vector addition rules or sign conventions.