Vectors Exercise 5.4 1. Let \( \mathbf{u}=(11,16) \) and \( \mathrm{v}=(-8,12) \). Find \( \begin{array}{llll}\begin{array}{ll}\text { a. } \mathbf{5 u} & \text { b. } \mathbf{3}(4 \mathbf{u})\end{array} \quad \text { c. }(5+\mathbf{3}) \mathbf{u} & \text { d. } \mathbf{4}(\mathbf{u}+\mathrm{v}) & \text { e. } 1 \cdot v \quad \mathrm{f} .0 \cdot \mathrm{v}\end{array} \) 2. In the triangle \( \mathrm{ABC}, \overrightarrow{A B} \) represents \( \mathbf{u} \) and \( \overrightarrow{B C} \) represents v . If D is the midpoint of AB , then express \( \overrightarrow{A C}, \overrightarrow{C A} \) and \( \overrightarrow{\mathrm{DC}} \) in terms of \( \mathbf{u} \) and v .

To find the required vector calculations, let's dive right in! For part \( a \), multiplying \( \mathbf{u} \) by 5 gives us \( 5\mathbf{u} = 5(11, 16) = (55, 80) \). For part \( b \), we first calculate \( 4\mathbf{u} = 4(11, 16) = (44, 64) \) and then \( 3(4\mathbf{u}) = 3(44, 64) = (132, 192) \). In part \( c \), we see \( 5 + 3 = 8 \), leading to \( (5 + 3)\mathbf{u} = 8\mathbf{u} = 8(11, 16) = (88, 128) \). For part \( d \), we have \( \mathbf{u} + \mathbf{v} = (11, 16) + (-8, 12) = (3, 28) \), and therefore \( \mathbf{4}(\mathbf{u} + \mathbf{v}) = 4(3, 28) = (12, 112) \). For part \( e \), calculating gives us \( 1\cdot\mathbf{v} = 1(-8, 12) = (-8, 12) \), and for part \( f \), \( 0\cdot\mathbf{v} = 0 \) leads to the zero vector, \( (0, 0) \). Now for the triangle \( ABC \)! The displacement \( \overrightarrow{A C} \) can be expressed as: \[ \overrightarrow{A C} = \overrightarrow{A B} + \overrightarrow{B C} = \mathbf{u} + \mathbf{v} \] For \( \overrightarrow{C A} \), it’s the negative of \( \overrightarrow{A C} \): \[ \overrightarrow{C A} = -(\mathbf{u} + \mathbf{v}) = -\mathbf{u} - \mathbf{v} \] Finally, for \( \overrightarrow{D C} \), where \( D \) is the midpoint of \( AB \): \[ \overrightarrow{D C} = \overrightarrow{D B} + \overrightarrow{B C} = \frac{1}{2} \mathbf{u} + \mathbf{v} \] That's how you express these vectors in terms of \( \mathbf{u} \) and \( \mathbf{v} \)! Happy vector calculating!