Question 31 of 68intermediate🔍 AnalyzeShort Answer2 marks

If a⃗ ⋅ a⃗ = 0 and a⃗ ⋅ b⃗ = 0, then what can be concluded about the vector b⃗?

Correct Answer

From the given condition a⃗ ⋅ a⃗ = 0, we can conclude that |a⃗|² = 0, which implies that a⃗ = 0⃗ (the zero vector).

Since a⃗ = 0⃗, the second condition a⃗ ⋅ b⃗ = 0 is automatically satisfied for any vector b⃗. This is because when either vector is the zero vector, the dot product is defined to be zero.

Therefore, nothing specific can be concluded about b⃗. The vector b⃗ can be any vector — it may be a zero vector or any non-zero vector. Both conditions are satisfied regardless of what b⃗ is.

Exercise: EXERCISE 10.3 | Q: 12 | (Page 25)
For More Understanding

Explanation

The question tests understanding of dot product properties. From the context: 'In particular, a⃗ ⋅ a⃗ = |a⃗|²'. So if a⃗ ⋅ a⃗ = 0, then |a⃗|² = 0, meaning a⃗ is the zero vector. The context also states: 'If either a⃗ = 0⃗ or b⃗ = 0⃗ then θ is not defined, and in this case, we define a⃗ ⋅ b⃗ = 0'. Since a⃗ is the zero vector, the dot product a⃗ ⋅ b⃗ equals zero for any b⃗. Hence, no specific conclusion can be drawn about b⃗.

Solution Steps

  1. Step 1: From a⃗ ⋅ a⃗ = 0, use the property that a⃗ ⋅ a⃗ = |a⃗|²

  2. Step 2: Conclude that |a⃗|² = 0, which means a⃗ = 0⃗ (zero vector)

  3. Step 3: Apply the rule that when a⃗ = 0⃗, the dot product a⃗ ⋅ b⃗ = 0 for any b⃗

  4. Step 4: Conclude that b⃗ can be any vector - no specific property can be determined

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