Question 43 of 68intermediate🔍 AnalyzeShort Answer2 marks

Given that a⃗ ⋅ b⃗ = 0 and a⃗ × b⃗ = 0⃗. What can you conclude about the vectors a⃗ and b⃗?

Correct Answer

From the given conditions, we can conclude that either a⃗=0⃗\vec{a} = \vec{0} or b⃗=0⃗\vec{b} = \vec{0} (or both are zero vectors).

Reasoning: For two nonzero vectors, a⃗×b⃗=0⃗\vec{a} \times \vec{b} = \vec{0} implies that the vectors are parallel to each other. On the other hand, a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0 implies that the vectors are perpendicular to each other.

It is impossible for two nonzero vectors to be both parallel and perpendicular simultaneously. Therefore, the only possibility is that at least one of the vectors must be a zero vector.

Exercise: EXERCISE 10.4 | Q: 6 | (Page 31)
For More Understanding

Explanation

This question tests the understanding of dot product and cross product properties. The key insight is that for nonzero vectors, the cross product being zero indicates parallel vectors, while the dot product being zero indicates perpendicular vectors. Since a vector cannot be both parallel and perpendicular to another nonzero vector simultaneously, the only solution is that at least one vector must be the zero vector. The context explicitly states that if either vector is zero, the cross product is defined as zero, and the dot product would also be zero.

Solution Steps

  1. Step 1: Analyze a⃗×b⃗=0⃗\vec{a} \times \vec{b} = \vec{0}: For nonzero vectors, this means vectors are parallel. If either vector is zero, cross product is also zero.

  2. Step 2: Analyze a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0: For nonzero vectors, this means vectors are perpendicular.

  3. Step 3: Combine both conditions: Nonzero vectors cannot be both parallel AND perpendicular simultaneously.

  4. Step 4: Conclude: At least one of the vectors must be the zero vector.

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