Education Blogs If 𝑎⃗ ≠ 0, ⃗⃗⃗ 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗, 𝑎⃗ ×...

If 𝑎⃗ ≠ 0, ⃗⃗⃗ 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗, 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗, then show that 𝑏⃗⃗ = 𝑐⃗.

CBSE Sample Question Paper, Class 12 Mathematics Term 2 Question - If 𝑎⃗ ≠ 0, ⃗⃗⃗ 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗, 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗, then show that 𝑏⃗⃗ = 𝑐⃗.

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May 18, 2022

If 𝑎⃗ ≠ 0,
⃗⃗⃗ 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗, 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗, then show that 𝑏⃗⃗ = 𝑐⃗.

Answer.

If 𝑎⃗ ≠ 0,
⃗⃗⃗ 𝑎⃗. 𝑏⃗⃗ = 𝑎⃗. 𝑐⃗, 𝑎⃗ × 𝑏⃗⃗ = 𝑎⃗ × 𝑐⃗, then show that 𝑏⃗⃗ = 𝑐⃗.
Solution: We have 𝑎⃗. (𝑏⃗⃗ − 𝑐⃗) = 0
⇒ (𝑏⃗⃗ − 𝑐⃗) = ⃗0⃗ or 𝑎⃗ ⊥ (𝑏⃗⃗ − 𝑐⃗)
⇒ 𝑏⃗⃗ = 𝑐⃗ or 𝑎⃗ ⊥ (𝑏⃗⃗ − 𝑐⃗)
Also, 𝑎⃗ × (𝑏⃗⃗ − 𝑐⃗) = ⃗0⃗
⇒ (𝑏⃗⃗ − 𝑐⃗) = ⃗0⃗ or 𝑎⃗ ∥ (𝑏⃗⃗ − 𝑐⃗)
⇒ 𝑏⃗⃗ = 𝑐⃗ or 𝑎⃗ ∥ (𝑏⃗⃗ − 𝑐⃗)
𝑎⃗ 𝑐𝑎𝑛 𝑛𝑜𝑡 𝑏𝑒 𝑏𝑜𝑡ℎ 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 (𝑏⃗⃗ − 𝑐⃗) 𝑎𝑛𝑑 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑡𝑜 (𝑏⃗⃗ − 𝑐⃗)
Hence, 𝑏⃗⃗ = 𝑐⃗.

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