Can Vectors be Added?

📌 CONCEPT: In vector algebra, vectors can be added together to form a new vector, but there are certain conditions under which the addition of two vectors may not result in a new vector.

📐 RULE / FORMULA: The addition of two vectors a and b is only possible if both vectors have the same initial point or if one vector is a zero vector. If the vectors have different initial points, we can translate one of the vectors so that they have the same initial point.

💡 WORKED EXAMPLE: Consider two vectors a = 3i + 4j and b = -2i + 5j. To add these vectors, we need to ensure they have the same initial point. We can translate vector b so that it has the same initial point as vector a. The sum of these vectors would be (3i + 4j) + (-2i + 5j) = (3 - 2)i + (4 + 5)j = i + 9j.

⚠️ COMMON MISTAKE: Students often assume that the addition of two vectors is always possible, regardless of their initial points. However, this is not true and requires careful consideration of the initial points of the vectors being added.