Scalar triple product practice · Mixed difficulty

The correct answer is D (9) because it is calculated using the shortest distance formula between two skew lines.

Option A (-42) is correct because given the conditions, the scalar triple product can be computed using the properties of coplanar vectors and perpendicularity, resulting in -42.

Option D ($\frac{95}{8}$) is the correct answer because it is the product of the roots of the equation obtained from the coplanarity condition.

Option D (27) is correct. The volume of the parallelepiped is given by the scalar triple product, which can be calculated using the determinant of the coefficients multiplied by the original scalar triple product.

Option C (-1) is correct because the condition for a vector to be parallel to the plane of two other vectors is correctly applied using the scalar triple product.

The correct answer is D ($- 1/4$) because the scalar triple product is zero, implying coplanarity, which leads to the solution λ = -1/4.

Option D (None of these) is correct because the condition for coplanarity leads to $s^2 = rt$, implying $s$ is the geometric mean of $r$ and $t$, but the options do not explicitly state this.

Option B ($\vec{{a}}-\vec{{b}}$) is correct because after expanding the expression $(\vec{{a}}+\vec{{b}}) \times (\vec{{a}} \times \vec{{b}})$, it simplifies to a vector parallel to $\vec{{a}}-\vec{{b}}$.

Option C ($\frac{[\mathbf{a} \mathbf{b} \mathbf{c}]}{\|\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}\|}$) is the correct length of the perpendicular from the origin to the plane because it uses the scalar triple product and the correct normal vector.

Option A is correct because the sum of the volumes of the tetrahedra formed by dropping perpendiculars from the origin to each face equals twice the volume of the tetrahedron.

Option C ($\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right] \vec{r}$) is correct because the given expression can be simplified using the properties of scalar triple products and vector decomposition.

Option B (9) is correct because by using the given conditions of perpendicularity and coplanarity, we can find the values of x, y, and z, and then calculate x² + y² + z².

Option A ($\alpha(\beta+\gamma)=\beta \gamma$) is correct because, by equating the magnitudes and simplifying, we arrive at the given relation.

Option B ($\frac{{16}}{{\sqrt{3}}}$) is correct because it is derived from the given condition a x (i + j + 2k) = 0 and |a|^2 = 8.

Option B (4) is correct because the volume of a parallelepiped is the absolute value of the scalar triple product of its sides.

Option D ($\left[\begin{array}{lll}\vec{b} & \vec{c} & \vec{a}\end{array}\right]$) is correct because the scalar triple product is invariant under cyclic permutations.

Option A ($\frac{1}{\sqrt{2}}$) is correct because the volume of the parallelepiped can be calculated using the determinant of the dot products of the unit vectors.

Option C ($1/\sqrt 3$) is correct because it minimizes the volume of the parallelepiped, found by taking the derivative of the volume with respect to 'a' and setting it to zero.

Option D (All of these) is correct because the condition for coplanarity leads to a+b+c = 0, implying that $\vec{v}$ is perpendicular to all three vectors.

Option C ($\frac{1}{3}(2\widehat i + 2\widehat j - \widehat k)$) is correct because the scalar triple product is maximized when 'a' is in the direction of b x c, and 'a' is a unit vector.

Option B (1) is correct because by using vector triple product and scalar triple product properties, the given expression simplifies to [a b c]², making lambda equal to 1.

Option B ($4 \sqrt{{3}}$) is correct because it is the shortest distance calculated using the formula for the shortest distance between two skew lines.

Option C (3) is correct because the scalar triple product simplifies to 3 after applying vector identities and given conditions.

Option B ($\mathbf{\vec a} \times ( {\mathbf{\vec b} + \mathbf{\vec c}} ) = \vec 0$) is correct because if $\mathbf{\vec a}$ is perpendicular to both $\mathbf{\vec b}$ and $\mathbf{\vec c}$, it is also perpendicular to their sum, and the cross product of perpendicular vectors is zero.

Option B is correct because it correctly expresses a complex number on the unit circle in terms of the tangent of half its argument.

Option A ($\pm 2$) is correct because the volume of the parallelepiped is calculated using the scalar triple product, leading to the equation $2c^2 = 8$.

Option C ($1 / \sqrt{3}$) is correct because the volume of the parallelepiped is minimized when the derivative of the volume with respect to 'a' is zero and the second derivative is positive, which occurs at $a = 1 / \sqrt{3}$.

Option D (0) is correct because the shortest distance between the lines is calculated to be 0, indicating that the lines intersect.

Option D ($-[{A}, {B}, {C}]$) is correct because expanding the triple scalar product and simplifying using properties of determinants yields this result.

Option C (non-zero) is correct because the scalar triple product of the inverse vectors is the inverse of the scalar triple product of the original vectors, and since the original is non-zero, the inverse is also non-zero.