1.If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors, prove that $[\vec{a}+\vec{b} \quad \vec{b}+\vec{c} \quad \vec{c}+\vec{a}] = 2[\vec{a} \, \vec{b} \, \vec{c}]$.
2.Prove Jacobi's Identity: $\vec{a} \times (\vec{b} \times \vec{c}) + \vec{b} \times (\vec{c} \times \vec{a}) + \vec{c} \times (\vec{a} \times \vec{b}) = \vec{0}$.
3.Let $\vec{a}$ be a unit vector and $\vec{b}, \vec{c}$ be non-collinear unit vectors. If $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b} + \vec{c}}{\sqrt{2}}$, find the angle that $\vec{a}$ makes with $\vec{b}$ and $\vec{c}$.
4.Solve the vector equation $\vec{x} + \vec{x} \times \vec{a} = \vec{b}$ for the unknown vector $\vec{x}$, in terms of vectors $\vec{a}$ and $\vec{b}$.
5.Prove the identity: $[\vec{a}\times\vec{b} \quad \vec{b}\times\vec{c} \quad \vec{c}\times\vec{a}] = [\vec{a} \, \vec{b} \, \vec{c}]^2$.
6.Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors such that $\vec{c}$ is a unit vector perpendicular to both $\vec{a}$ and $\vec{b}$. If the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{6}$, prove that $[\vec{a} \, \vec{b} \, \vec{c}]^2 = \frac{1}{4}|\vec{a}|^2|\vec{b}|^2$.
7.Find a vector $\vec{v}$ of magnitude $\sqrt{8}$ which is coplanar with $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k}$, and is orthogonal to $\vec{c} = \hat{i} + 2\hat{j} - \hat{k}$.
8.If $\vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors of equal magnitude, find the angle between $\vec{a}$ and the vector $(\vec{a} + \vec{b} + \vec{c})$.
9.Using vector methods, prove that the altitudes of a triangle are concurrent (intersect at the orthocenter).
10.Using vector methods, prove that the angle inscribed in a semicircle is a right angle.
11.State and mathematically prove the Cauchy-Schwarz inequality for any two vectors $\vec{a}$ and $\vec{b}$: $|\vec{a} \cdot \vec{b}| \le |\vec{a}| |\vec{b}|$.
12.If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors satisfying $|\vec{a}-\vec{b}|^2 + |\vec{b}-\vec{c}|^2 + |\vec{c}-\vec{a}|^2 = 9$, prove that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$.
13.The volume of a tetrahedron with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $V$. Prove that the volume of the tetrahedron with coterminous edges $\vec{a}+\vec{b}, \vec{b}+\vec{c}, \vec{c}+\vec{a}$ is $2V$.
14.Given $\vec{a} = x\hat{i} + 12\hat{j} - \hat{k}$, $\vec{b} = 2\hat{i} + 2x\hat{j} + \hat{k}$, and $\vec{c} = \hat{i} + \hat{k}$. Find the value of $x$ for which the scalar triple product $[\vec{a} \, \vec{b} \, \vec{c}]$ is minimum.
15.Let $\vec{a}$ and $\vec{b}$ be two unit vectors. If the vectors $\vec{c} = \vec{a} + 2\vec{b}$ and $\vec{d} = 5\vec{a} - 4\vec{b}$ are perpendicular to each other, find the angle between $\vec{a}$ and $\vec{b}$.
16.Let $\vec{p}, \vec{q}, \vec{r}$ represent the reciprocal system of vectors to the non-coplanar vectors $\vec{a}, \vec{b}, \vec{c}$. Prove that $\vec{a} \cdot \vec{p} + \vec{b} \cdot \vec{q} + \vec{c} \cdot \vec{r} = 3$.
17.Prove the identity: $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c})$.
18.Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{b} \times \vec{c} = \vec{a}$. Prove that $\vec{a}, \vec{b}, \vec{c}$ are mutually orthogonal, and find the magnitude of $\vec{b}$.
19.If $\hat{a}$ and $\hat{b}$ are unit vectors inclined at an angle $\theta$, prove that $\sin\left(\frac{\theta}{2}\right) = \frac{1}{2}|\hat{a} - \hat{b}|$.
20.Find the value of $y$ if the four points $A(1,1,1)$, $B(2,0,1)$, $C(1,2,0)$, and $D(3, y, 2)$ are coplanar.
21.Prove the identity: $[\vec{a}-\vec{b} \quad \vec{b}-\vec{c} \quad \vec{c}-\vec{a}] = 0$.
22.If the vectors $x\hat{i} - \hat{j} + \hat{k}$, $2\hat{i} + y\hat{j} - z\hat{k}$ and $3\hat{i} - \hat{j} + 2\hat{k}$ form an orthogonal basis in 3D space, find the values of $x, y,$ and $z$.
23.Using vector methods, prove that the medians of a triangle are concurrent (intersect at the centroid) and divide each other in the ratio 2:1.
24.The diagonals of a parallelogram are given by $\vec{d_1} = 3\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{d_2} = \hat{i} - 3\hat{j} + 4\hat{k}$. Find the lengths of the adjacent sides of the parallelogram.
25.A particle acted upon by constant forces $\vec{F_1} = 4\hat{i} + \hat{j} - 3\hat{k}$ and $\vec{F_2} = 3\hat{i} + \hat{j} - \hat{k}$ is displaced from the point $P(\hat{i} + 2\hat{j} + 3\hat{k})$ to the point $Q(5\hat{i} + 4\hat{j} + \hat{k})$. Find the total work done.
26.Determine a unit vector perpendicular to the plane containing the vectors $\vec{a} = 2\hat{i} - 6\hat{j} - 3\hat{k}$ and $\vec{b} = 4\hat{i} + 3\hat{j} - \hat{k}$.
27.Prove that the area of a triangle with position vertices $\vec{a}, \vec{b}, \vec{c}$ is given by $\frac{1}{2} |\vec{a}\times\vec{b} + \vec{b}\times\vec{c} + \vec{c}\times\vec{a}|$.
28.If $\vec{a}, \vec{b}, \vec{c}$ are coplanar vectors, prove mathematically that $\vec{a}\times\vec{b}$, $\vec{b}\times\vec{c}$, and $\vec{c}\times\vec{a}$ are also coplanar vectors.
29.If $\vec{a} \times \vec{b} = \vec{c} \times \vec{d}$ and $\vec{a} \times \vec{c} = \vec{b} \times \vec{d}$, prove that the vector $(\vec{a}-\vec{d})$ is parallel to the vector $(\vec{b}-\vec{c})$, assuming $\vec{a} \neq \vec{d}$ and $\vec{b} \neq \vec{c}$.
30.The volume of a parallelepiped formed by the vectors $\vec{a}\times\vec{b}, \vec{b}\times\vec{c}$, and $\vec{c}\times\vec{a}$ is $36$ cubic units. What is the volume of the parallelepiped formed by the vectors $\vec{a}, \vec{b}, \vec{c}$?
