If is a vector and m is a scalar such that m \(\vec a\) = \(\vec 0\),

1.	If is a vector and m is a scalar such that m \(\vec a\) = \(\vec 0\), then what are the alternatives for m and \(\vec a\)?
Answer» Given \(\vec a\)is a vector and m is a scalar such that m\(\vec a\) = \(\vec 0\) Let \(\vec a\) = \(a_1\hat i+b_1\hat j + c_1\hat k\) then according to the given question m\(\vec a\) = \(\vec 0\) ⇒ \(m(a_1\hat i+b_1\hat j+c_1\hat k)\) = \(0\hat i+0\hat j+0\hat k\) ⇒ \((ma_1\hat i+mb_1\hat j+mc_1\hat k)\) = \(0\hat i+0\hat j+0\hat k\) Compare the coefficients of \(\hat i, \hat j,\hat k\), we get ma₁ = 0 ⇒ m = 0 or a₁ = 0 Similarly, mb₁ = 0 ⇒ m = 0 or b₁ = 0 And, mc₁ = 0 ⇒ m = 0 or c₁ = 0 From the above three conditions, m = 0 or a₁ = b₁ = c₁ = 0 ⇒ m = 0 or \(\vec a\) = \(a_1\hat i+b_1\hat j + c_1\hat k\) = \(0\hat i+0\hat j+0\hat k\) = 0 Hence the alternatives for m and \(\vec a\) are m = 0 or \(\vec a\) = 0

If is a vector and m is a scalar such that m \(\vec a\) = \(\vec 0\), then what are the alternatives for m and \(\vec a\)?

Answer»

Given \(\vec a\)is a vector and m is a scalar such that m\(\vec a\) = \(\vec 0\)

Let \(\vec a\) = \(a_1\hat i+b_1\hat j + c_1\hat k\)

then according to the given question

m\(\vec a\) = \(\vec 0\)

⇒ \(m(a_1\hat i+b_1\hat j+c_1\hat k)\) = \(0\hat i+0\hat j+0\hat k\)

⇒ \((ma_1\hat i+mb_1\hat j+mc_1\hat k)\) = \(0\hat i+0\hat j+0\hat k\)

Compare the coefficients of \(\hat i, \hat j,\hat k\), we get

ma₁ = 0 ⇒ m = 0 or a₁ = 0

Similarly, mb₁ = 0 ⇒ m = 0 or b₁ = 0

And, mc₁ = 0 ⇒ m = 0 or c₁ = 0

From the above three conditions,

m = 0 or a₁ = b₁ = c₁ = 0

⇒ m = 0 or \(\vec a\) = \(a_1\hat i+b_1\hat j + c_1\hat k\) = \(0\hat i+0\hat j+0\hat k\) = 0

Hence the alternatives for m and \(\vec a\) are m = 0 or \(\vec a\) = 0

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