Question

Let \(3 \mathbf{i}-\mathbf{j}+4 \mathbf{k}, 7 \mathbf{j}-2 \mathbf{k}, \mathbf{i}-3 \mathbf{j}+\mathbf{k}\) be three vectors with tails at the origin. Then their heads determine three points \(A, B, C\) in space which form a triangle. Find vectors representing the sides \(A B, B C, C A\) in that order and direction (for example, \(A\) to \(B,\) not \(B\) to \(A\) ) and show that the sum of these vectors is zero.

Short answer

Vectors AB: \(-3i + 8j - 6k\), Vector BC: \(i - 10j + 3k\), Vector CA: \(2i + 2j + 3k\). Sum is zero.

Step-by-step solution

Identifying the Points

The points A, B, and C are determined by the heads of the vectors: Point A is at \((3i - j + 4k)\), point B is at \((7j - 2k)\), and point C is at \((i - 3j + k)\).

Calculating Vector AB

Vector AB is found by subtracting the coordinates of A from B: \( \text{AB} = (7j - 2k) - (3i - j + 4k) = -3i + 8j - 6k \)

Calculating Vector BC

Vector BC is found by subtracting coordinates of B from C: \( \text{BC} = (i - 3j + k) - (7j - 2k) = i - 10j + 3k \)

Calculating Vector CA

Vector CA is found by subtracting the coordinates of C from A: \( \text{CA} = (3i - j + 4k) - (i - 3j + k) = 2i + 2j + 3k \)

Sum of Vectors AB, BC, and CA

Sum the vectors AB, BC, and CA: \( AB + BC + CA = (-3i + 8j - 6k) + (i - 10j + 3k) + (2i + 2j + 3k)= (-3i + i + 2i) + (8j - 10j + 2j) + (-6k + 3k + 3k) = 0 \). Therefore, the sum of the vectors AB, BC, and CA is zero, completing the triangle.

Key concepts

Vector Subtraction

Vector subtraction is key to determining the sides of a triangle formed by vectors.
It's like finding the difference between two points.
To subtract vectors, you subtract the corresponding components.
For example, to find vector AB from points A and B:
You have Point A at \((3\bm{i} - \bm{j} + 4\bm{k})\) and Point B at \((7\bm{j} - 2\bm{k})\).
Subtract each part of A from B:

• i-component: 0 - 3 = -3
• j-component: 7 - (-1)= 8
• k-component: -2 - 4 = -6

The result is vector AB = \(-3\bm{i} + 8\bm{j} - 6\bm{k}\).

Keep this method in mind for subtracting vectors in other problems too.

Vector Coordinates

Vector coordinates tell us the position of the vector in space.
In our exercise, each vector mentions points A, B, and C with origin as the tail.
Here, Point A corresponds to the vector \((3\bm{i} - \bm{j} + 4\bm{k})\).
This means A has coordinates (3, -1, 4).

• 3 is the i-component
• -1 is the j-component
• 4 is the k-component.

Similarly, Point B corresponds to the vector \((7\bm{j} - 2\bm{k})\).
Note that B has coordinates (0, 7, -2).
Finally, Point C corresponds to the vector \((\bm{i} - 3\bm{j} + \bm{k})\).
C has coordinates (1, -3, 1).

Understanding these coordinates helps visualize the vector's position.

Vector Sum

Vector sum is the process of adding vectors.
In our example, we need to add vectors AB, BC, and CA.
Sum each component of the vectors separately:

• For the i-components: \(-3 + 1 + 2 = 0\)
• For the j-components: \(8 - 10 + 2 = 0\)
• For the k-components: \(-6 + 3 + 3 = 0\)

The vector sum is \((0\bm{i} + 0\bm{j} + 0\bm{k})\).
This result confirms the sum of those vectors is zero.
It is always good practice to check your computations step by step.

Adding vectors correctly is vital for solving multiple physics and math problems.

Properties of Triangles

Understanding properties of triangles helps to solve vector problems.
When three vectors form a triangle, their sum is zero.
This is because each sequence of vectors cancels out.
For example, vector AB + BC + CA = 0, confirming they form a closed triangle in space.

• Triangle Inequality theorem: The sum of the lengths of any two sides must be greater than the length of the third side.
• A triangle has internal angles which add up to 180 degrees.
• The position vectors are invaluable in understanding the shape and size of the triangle.

Remember, these properties come in handy while analyzing shapes and figures in various math problems.