For the vectors \(\rm \vec a = -4\hat i + 2\hat j\), \(\rm \vec b =2\hat i + \hat j\) and \(\rm \vec c = 2\hat i + 3\hat j\), if \(\rm \vec c = m\vec a + n\vec b\), then the value of m + n is:

Concept:

If two vectors \(\rm \vec a = {a_1}\hat i + {a_2}\hat j+{a_3}\hat k\) and \(\rm \vec b = {b_1}\hat i + {b_2}\hat j+{b_3}\hat k\) are equal, then a₁ = b₁, a₂ = b₂ and c₁ = c₂.

Calculation:

We have \(\rm \vec c = m\vec a + n\vec b\).

⇒ 2î + 3ĵ = m(-4î + 2ĵ) + n(2î + ĵ)

⇒ 2î + 3ĵ = (-4m + 2n)î + (2m + n)ĵ

Equating the scalar coefficients, we get:

-4m + 2n = 2               ... (1)

2m + n = 3               ... (2)

Multiplying equation (2) by 2 and adding to equation (1), we get:

4n = 8

⇒ n = 2

Using either of the equations above, we also get:

m = \(\frac12\)

∴ m + n = 2 + \(\frac12\) = \(\frac{5}{2}\).