Question

select the correct answer from each drop - down menu.
if vector \\( \mathbf{u} = (5, -7) \\) and \\( \mathbf{v} = (-11, 3) \\), \\( 2\mathbf{v} - 6\mathbf{u} = \\) \\( \boldsymbol{\square} \\) and \\( \\| 2\mathbf{v} - 6\mathbf{u}\\| \approx \\) \\( \boldsymbol{\square} \\).

Explanation:

Step1: Calculate 2v

To find \(2\mathbf{v}\), we multiply each component of \(\mathbf{v}=(-11, 3)\) by 2. So, \(2\mathbf{v}=2\times(-11, 3)=(-22, 6)\).

Step2: Calculate 6u

To find \(6\mathbf{u}\), we multiply each component of \(\mathbf{u}=(5, -7)\) by 6. So, \(6\mathbf{u}=6\times(5, -7)=(30, -42)\).

Step3: Calculate \(2\mathbf{v}-6\mathbf{u}\)

Subtract the corresponding components of \(6\mathbf{u}\) from \(2\mathbf{v}\). So, \(2\mathbf{v}-6\mathbf{u}=(-22 - 30, 6 - (-42))=(-52, 48)\).

Step4: Calculate \(\|2\mathbf{v}-6\mathbf{u}\|\)

The magnitude of a vector \((x, y)\) is given by \(\sqrt{x^{2}+y^{2}}\). For the vector \((-52, 48)\), we have \(\|2\mathbf{v}-6\mathbf{u}\|=\sqrt{(-52)^{2}+48^{2}}=\sqrt{2704 + 2304}=\sqrt{5008}\approx70.77\) (or we can simplify \(\sqrt{5008}=\sqrt{16\times313}=4\sqrt{313}\approx70.77\)).

Answer:

First drop - down: \((-52, 48)\)
Second drop - down: Approximately \(70.77\) (or \(4\sqrt{313}\) or the exact value of \(\sqrt{5008}\))