Question 33 of 68intermediate💡 UnderstandShort Answer2 marks

If either vector a⃗ = 0⃗ or b⃗ = 0⃗, then a⃗ ⋅ b⃗ = 0. But the converse need not be true. Justify your answer with an example.

Correct Answer

If either a⃗=0⃗\vec{a} = \vec{0} or b⃗=0⃗\vec{b} = \vec{0}, then θ\theta is not defined, and in this case, we define a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0. Hence, the statement is true.

However, the converse need not be true. According to the observations, if a⃗\vec{a} and b⃗\vec{b} are two nonzero vectors that are perpendicular to each other, then a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0.

Example: Consider a⃗=i^\vec{a} = \hat{i} and b⃗=j^\vec{b} = \hat{j}. Both are non-zero vectors, yet a⃗⋅b⃗=i^⋅j^=0\vec{a} \cdot \vec{b} = \hat{i} \cdot \hat{j} = 0 because i^\hat{i} and j^\hat{j} are perpendicular to each other.

Thus, a⃗⋅b⃗=0\vec{a} \cdot \vec{b} = 0 does not imply that either a⃗=0⃗\vec{a} = \vec{0} or b⃗=0⃗\vec{b} = \vec{0}.

Exercise: EXERCISE 10.3 | Q: 14 | (Page 25)
For More Understanding

Explanation

The question tests understanding of the dot product property. From the textbook context: (1) If either vector is zero, the dot product is defined to be zero. (2) For two nonzero vectors, the dot product equals zero if and only if they are perpendicular. This means the converse is false - zero dot product can occur even when both vectors are nonzero, provided they are perpendicular. The example using unit vectors î and ĵ directly from the textbook context demonstrates this clearly.

Solution Steps

  1. Step 1: State that if either vector is zero, dot product is defined as zero.

  2. Step 2: Explain that for nonzero vectors, dot product is zero when vectors are perpendicular.

  3. Step 3: Provide example: a⃗ = î, b⃗ = ĵ (both nonzero but perpendicular).

  4. Step 4: Show î ⋅ ĵ = 0, proving converse is not true.

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