Scalar triple product practice · Mixed difficulty

Option C (16) is correct because the determinant of the matrix formed by dot products is equal to the square of the scalar triple product.

Option A (3) is correct because each term in the expression equals 1, and there are three terms.

Option B (1) is correct because the equation derived from the coplanarity condition has only one distinct real solution, which is λ = 0.

Option A (0) is correct because linearly dependent vectors have a scalar triple product of zero, leading to lambda = 0.

Option A ($\left.\mid \begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right] \mid$) is correct because the expression simplifies to the absolute value of the scalar triple product.

Option B (10 cu units) is correct because it is the volume of the parallelepiped calculated using the scalar triple product of the vectors formed by the given vertices.

Option D ($-3\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right]$) is the correct answer because the scalar triple product can be evaluated using determinant properties.

Option B (1) is correct because by taking the dot product of \vec{r} with (\vec{a} + \vec{b} + \vec{c}) and simplifying using scalar triple product properties, we find that [\vec{a} \vec{b} \vec{c}] = 1.

Option B ($\mathbf{\vec b} \times \mathbf{\vec c} + 2\mathbf{\vec c} \times \mathbf{\vec a} + 3\mathbf{\vec a} \times \mathbf{\vec b}$) is correct because it represents the vector $\mathbf{\vec r}$ as a linear combination of cross products of the given non-coplanar vectors.

Option C (2) is correct because the determinant equation leads to ${\left( {1 + {\lambda ^2}} \right)^2}\left( {{\lambda ^2} - 2} \right) = 0$, giving two distinct real values for $\lambda$.

Option D (5) is the correct answer because it results in a volume of 11 cubic units for the tetrahedron.

Option A ($l^2+m^2+n^2$) is correct because the dot product simplifies to the sum of the squares of the coefficients due to the properties of reciprocal vectors.

The correct answer is D (None of these) because the expression evaluates to 0 since the vectors are coplanar.

Option C ($2[\vec{{a}} \vec{{b}} \vec{{c}}]$) is incorrect. The correct answer is B ($[\vec{{a}} \vec{{b}} \vec{{c}}]$) because after expanding the scalar triple product, the result simplifies to the scalar triple product of a, b, and c.

Option B (0) is correct because the sum of the distinct real values of $\mu$ for which the given vectors are coplanar is 0.

The correct answer is B ($7{[ab\;c]^2}$) because it is obtained by expanding the scalar triple product and using the properties of determinants to simplify the expression.

Option A ($\mathbf{a}$) is correct because the expression represents the vector $\mathbf{a}$ decomposed into its components along the i, j, and k axes, and then reconstructed.

Option A (3) is correct because the shortest distance between the lines is given as $\frac{1}{\sqrt{38}}$, and solving for $\lambda$ using the formula for shortest distance yields $\lambda = 3$.

Option D (-5) is correct because by simplifying the given expression using vector identities and the provided values for $\vec{a}$ and $\vec{b}$, the result is -5.

Option D ($\frac{{5 \pi^2}}{4}$) is correct because it is the minimum value of x² + y² that satisfies the given conditions.

Option C ($\vec 0$) is correct because the sum of the cross products of reciprocal vectors is equal to the zero vector.

Option C (${\left[ {\;\mathbf{\vec a}\;\mathbf{\vec b}\;\mathbf{\vec c}\; } \right]^2}$) is correct because the scalar triple product of cross products simplifies to the square of the original scalar triple product.

Option C (7 unit) is correct because the volume of a parallelepiped is found by calculating the scalar triple product of its sides, which equals 7.

Option A ($\frac{\vec{a} \times(\vec{c} \times \vec{b})}{\vec{a} \cdot \vec{b}}$) is correct because by using the given conditions and vector identities, we can express $\vec{r}$ in terms of $\vec{a}$, $\vec{b}$, and $\vec{c}$.

The correct answer is B (2) because there are two real values of 'a' for which the given vectors are coplanar, obtained by setting the determinant of the vectors to zero and solving the resulting quadratic equation.

Option C (no value of $\lambda$) is correct because the equation simplifies to $\lambda^4 = -1$, which has no real solutions for $\lambda$.

Option A (z\widehat i - x\hat k$z\widehat \mathbf{i} - x\mathbf{\hat k}$) is correct because in a right-handed system, the cross product of \vec{b} and \vec{a} gives \vec{c}.

Option D (None of these) is correct because the volume of the cube, calculated using the scalar triple product, is 3696, which is not among the given options.

Option D ($\mathbf{a} \cdot \mathbf{b}=\mathbf{b} \cdot \mathbf{c}=\mathbf{c} \cdot \mathbf{a}=0$) is correct because it implies that the vectors are mutually perpendicular, satisfying the given condition.

Option B (1) is correct because the given expression simplifies to $\left[\begin{array}{lll}\vec{a} & \vec{b} & \vec{c}\end{array}\right]^2$, making $\lambda = 1$.