good ! your answer is correctgreg1313 said:\(\displaystyle \vec{AC}\cdot\vec{AD}=|\vec{AC}||\vec{AD}|\cos(60)=\frac12|\vec{AC}||\vec{AD}|=|\vec{AC}|\implies|\vec{AD}|=2\)
\(\displaystyle |\vec{AC}|\cos(60)=2\implies|\vec{AC}|=4\implies|\vec{AB}|=\sqrt{12}\)
\(\displaystyle \vec{AC}\cdot\vec{AB}=4\sqrt{12}\cos(30)=12\)
This phrase is asking for the dot product of the vectors $\vec{AC}$ and $\vec{AB}$ in the context of a rectangle with vertices $A$, $B$, $C$, and $D$. The dot product is a mathematical operation that returns a scalar value representing the projection of one vector onto another.
A vector is a quantity that has both magnitude and direction. In mathematics and physics, vectors are often represented by arrows and can be written in terms of their components, such as $\vec{v} = (v_x, v_y)$.
The dot product of two vectors, $\vec{a}$ and $\vec{b}$, is equal to the product of their magnitudes multiplied by the cosine of the angle between them. In mathematical notation, this can be written as $\vec{a} \cdot \vec{b} = \lvert \vec{a} \rvert \lvert \vec{b} \rvert \cos\theta$, where $\theta$ is the angle between the two vectors.
In a rectangle, the dot product of the vectors $\vec{AC}$ and $\vec{AB}$ can tell us about the relationship between the sides and angles of the rectangle. For example, if the dot product is zero, the vectors are perpendicular and the rectangle is a right angle. If the dot product is positive, the vectors are acute and the rectangle is an acute angle. If the dot product is negative, the vectors are obtuse and the rectangle is an obtuse angle.
Yes, the dot product is used in many areas of science, including physics, engineering, and computer graphics. It is used to calculate work, determine the angle between two vectors, and find the projection of one vector onto another. It also has applications in calculus and linear algebra.