Understanding the Relationship Between Vectors: Perpendicularity and Magnitude

In vector algebra, understanding the relationship between vectors, their perpendicularity, and magnitudes is fundamental to solving complex geometric and physical problems. This article delves into a specific scenario where the result of two vectors a and b is perpendicular to vector a and its magnitude is equal to half of the magnitude of vector b. We will explore the necessary steps to determine the angle between these vectors using vector properties and the relationship between their magnitudes.

Step-by-Step Solution

Let's denote vector a as a, vector b as b, and the resultant vector c a ? b. We are given two key pieces of information:

The vector c is perpendicular to a. The magnitude of c is equal to half the magnitude of b: c 1/2 b.

Step 1: Use the Perpendicular Condition

Since c is perpendicular to a, we have:

a ? c 0

Substituting c a ? b into the dot product:

a ? a ? b 0

This simplifies to:

a2 - a ? b 0

Step 2: Rearranging the Dot Product

Rearranging the equation from step 1, we find:

a ? b -a2

Step 3: Relate the Magnitudes

The dot product a ? b can also be expressed in terms of the magnitudes of the vectors and the cosine of the angle θ between them:

a ? b |a| |b| cos(θ)

Setting this equal to our result from step 2:

|a| |b| cos(θ) -a2

Step 4: Dividing by a

Assuming a ≠ 0, we can divide both sides by a to get:

|b| cos(θ) -|a|

Step 5: Finding the Magnitude of c

We know that c 1/2 b. Therefore, we can express c in terms of a and b:

c a ? b |a| |b| cos(θ) -|a|2

Substituting a ? b -|a|2 into the expression for c gives:

c -|a| |b| cos(θ) -|a|2

This can be rewritten as:

|c| |b| cos(θ) (1/2) |b|

Given that |c| (1/2) |b, we have:

|b| cos(θ) (1/2) |b|

Dividing by |b| (assuming b ≠ 0), we get:

cos(θ) -1/2

Step 6: Finding the Angle θ

The angle θ corresponding to cos(θ) -1/2 is either 150° or 210°.

Conclusion

Thus, the angle between vector a and vector b is either 150° or 210°.

By exploring the relationship between vectors, their perpendicularity, and magnitudes, we have successfully determined the possible angles between vectors a and b. This problem not only strengthens our understanding of vector algebra but also highlights the importance of the dot product in solving geometric and physical problems.