Question

determine the dot or scalar product of the pairs of vectors shown in the figures. the magnitudes of the vectors are a = 5.00, b = 6.00, and d = 7.00, and the angles are as given.
(a) \\(\vec{a} \cdot \vec{b}\\)
(b) \\(\vec{a} \cdot \vec{d}\\)

Explanation:

Step1: Find angle for $\vec{A} \cdot \vec{B}$

The angle between $\vec{A}$ and $\vec{B}$ is $60^\circ + 30^\circ = 90^\circ$.

Step2: Calculate $\vec{A} \cdot \vec{B}$

Use dot product formula $\vec{A} \cdot \vec{B} = |A||B|\cos\theta$
$\vec{A} \cdot \vec{B} = 5.00 \times 6.00 \times \cos(90^\circ)$
$\cos(90^\circ)=0$, so $\vec{A} \cdot \vec{B} = 5.00 \times 6.00 \times 0 = 0$

Step3: Find angle for $\vec{A} \cdot \vec{D}$

The angle between $\vec{A}$ and $\vec{D}$ is $180^\circ - 60^\circ - 60^\circ = 60^\circ$.

Step4: Calculate $\vec{A} \cdot \vec{D}$

Use dot product formula $\vec{A} \cdot \vec{D} = |A||D|\cos\theta$
$\vec{A} \cdot \vec{D} = 5.00 \times 7.00 \times \cos(60^\circ)$
$\cos(60^\circ)=0.5$, so $\vec{A} \cdot \vec{D} = 5.00 \times 7.00 \times 0.5 = 17.5$

Answer:

(a) $0$
(b) $17.5$