MAT 103 - Elementary Mathematics III Mock Exam 1

Given a scalar function $\phi(t) = t^2$ and a vector function $A(t) = t i + j$. Find $\frac{d}{dt}(\phi A)$.

Given $A = i + 2j + 3k$, $B = 2i - 3j + k$, and $C = 3i - j + 4k$. Find $C \cdot (A \times B)$.

If $r(t) = (t^3)i - (2t)j + (\ln t)k$, calculate $\frac{d^2r}{dt^2}$ at $t=1$.

Given two vectors $\vec{P}$ and $\vec{Q}$ such that $|\vec{P}| = 5$, $|\vec{Q}| = 6$, and $\vec{P} \cdot \vec{Q} = 15$. Find the cosine of the angle between $\vec{P}$ and $\vec{Q}$.

The cross product is anti-commutative, meaning $\mathbf{a} \times \mathbf{b}$ is equal to:

Let $U = (4, -5)$ and $V = (-2, -3)$. Find the magnitude of the vector $V - U$, denoted as $|V - U|$.

Given $A = i + 2j + 3k$, $B = 2i - 3j + k$, and $C = 3i + j - 2k$. Find $C \cdot (A \times B)$.

Which of the following statements is true for the cross product of two vectors $\mathbf{a}$ and $\mathbf{b}$?

Find the modulus of the vector $2i + 3j - 4k$.

Determine the cross product of $\mathbf{u} = (1, 2, 0)$ and $\mathbf{v} = (3, 4, 0)$.