Question

question
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problem 1: vector \\( \mathbf{w} \\) has initial point \\( (-2, -1) \\) and terminal point \\( (3, -5) \\). write \\( \mathbf{w} \\) as a linear combination of the standard unit vectors \\( \mathbf{i} \\) and \\( \mathbf{j} \\).
\\( \mathbf{w} = \\)
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Explanation:

Step1: Find the component form of the vector

To find the component form of a vector with initial point \((x_1, y_1)\) and terminal point \((x_2, y_2)\), we use the formula \(\vec{v}=\langle x_2 - x_1, y_2 - y_1
angle\). Here, \(x_1=-2\), \(y_1 = - 1\), \(x_2=3\), and \(y_2=-5\). So, the \(x\)-component is \(3-(-2)=3 + 2=5\) and the \(y\)-component is \(-5-(-1)=-5 + 1=-4\).

Step2: Write as a linear combination of \(\mathbf{i}\) and \(\mathbf{j}\)

A vector \(\langle a,b
angle\) can be written as \(a\mathbf{i}+b\mathbf{j}\). So, the vector \(\vec{w}\) with components \(\langle5,-4
angle\) is \(5\mathbf{i}-4\mathbf{j}\).

Answer:

\(5\mathbf{i}-4\mathbf{j}\)