Question

The vector (a) is a multiple of the vector (2i +3j) and (b) is a multiple of (2i+5j) The sum (a+b) is a multiple of the vector (8i +15j). Given that /a+b/= 34 and the scaler multiple of (8i+15j) is positive, Find the magnitude of a and b.​

Answer 1

`\|a\| = 5√(13)`.

`\|b\| = 3√(29)`.

Explanation:

Let
`m`,
`n`, and
`k` be scalars such that:

`\displaystyle a = m\, (2\, \vec{i} + 3\, \vec{j}) = m\, \begin{bmatrix}2 \\ 3\end{bmatrix}`.

`\displaystyle b = n\, (2\, \vec{i} + 5\, \vec{j}) = n\, \begin{bmatrix}2 \\ 5\end{bmatrix}`.

`\displaystyle (a + b) = k\, (8\, \vec{i} + 15\, \vec{j}) = k\, \begin{bmatrix}8 \\ 15\end{bmatrix}`.

The question states that
`\| a + b \| = 34`. In other words:

`k\, \sqrt{8^(2) + 15^(2)} = 34`.

`k^(2) \, (8^(2) + 15^(2)) = 34^(2)`.

`289\, k^(2) = 34^(2)`.

Make use of the fact that
`289 = 17^(2)` whereas
`34 = 2 * 17`.

`\begin{aligned}17^(2)\, k^(2) &= 34^(2)\\ &= (2 * 17)^(2) \\ &= 2^(2) * 17^(2) \end{aligned}`.

`k^(2) = 2^(2)`.

The question also states that the scalar multiple here is positive. Hence,
`k = 2`.

Therefore:

`\begin{aligned} (a + b) &= k\, (8\, \vec{i} + 15\, \vec{j}) \\ &= 2\, (8\, \vec{i} + 15\, \vec{j}) \\ &= 16\, \vec{i} + 30\, \vec{j}\\ &= \begin{bmatrix}16 \\ 30 \end{bmatrix}\end{aligned}`.

`(a + b)` could also be expressed in terms of
`m` and
`n`:

`\begin{aligned} a + b &= m\, (2\, \vec{i} + 3\, \vec{j}) + n\, (2\, \vec{i} + 5\, \vec{j}) \\ &= (2\, m + 2\, n) \, \vec{i} + (3\, m + 5\, n) \, \vec{j} \end{aligned}`.

`\begin{aligned} a + b &= m\, \begin{bmatrix}2\\ 3 \end{bmatrix} + n\, \begin{bmatrix} 2\\ 5 \end{bmatrix} \\ &= \begin{bmatrix}2\, m + 2\, n \\ 3\, m + 5\, n\end{bmatrix}\end{aligned}`.

Equate the two expressions and solve for
`m` and
`n`:

`\begin{cases}2\, m + 2\, n = 16 \\ 3\, m + 5\, n = 30\end{cases}`.

`\begin{cases}m = 5 \\ n = 3\end{cases}`.

Hence:

`\begin{aligned} \| a \| &= \| m\, (2\, \vec{i} + 3\, \vec{j})\| \\ &= m\, \| (2\, \vec{i} + 3\, \vec{j}) \| \\ &= 5\, \sqrt{2^(2) + 3^(2)} = 5 √(13)\end{aligned}`.

`\begin{aligned} \| b \| &= \| n\, (2\, \vec{i} + 5\, \vec{j})\| \\ &= n\, \| (2\, \vec{i} + 5\, \vec{j}) \| \\ &= 3\, \sqrt{2^(2) + 5^(2)} = 3 √(29)\end{aligned}`.