Addition of vectors practice · Mixed difficulty

Option D (Any real number) is correct because the collinearity condition leads to an equation where k can take any real value.

Option C (are collinear) is correct because the vectors AB and BC are scalar multiples of each other, indicating that the points A, B, and C lie on the same line.

Option B ($\frac{1}{3}$) is correct because for the vectors to be collinear, one must be a scalar multiple of the other, leading to a solvable equation for x.

Option A ($- 2\widehat i - 4\widehat j + 2\widehat k$) is correct because adding this vector to the given vector results in the unit vector $\widehat i$.

Option C is correct. Statement A is true because the magnitude of a vector is always greater than or equal to the magnitude of any of its components. Statement B is also true because the magnitude of a vector is less than or equal to 3 times the maximum of the absolute values of its components.

Option C ($\cos ^{-1} \frac{1}{\sqrt{3}}$) is correct because it is the angle between the vectors calculated using the dot product and magnitudes of the vectors.

Option D ($\frac{\mathbf{i}+\mathbf{j}}{\sqrt{2}}$) is correct because it represents a unit vector perpendicular to the plane containing the given vectors, found by normalizing their cross product.

Option C ($-\mathbf{i}+\mathbf{j}$) is correct because it represents the position vector of the midpoint C, calculated from the coordinates of points A and B.

Option B ($3 < x < 7$) is correct because it represents the solution set for the inequality after simplification and solving for x.

Option A (Linearly dependent) is correct because in a 3-dimensional space, any set of four or more vectors is always linearly dependent.

Option D ($\mathbf{\vec a} - 3\mathbf{\vec b}$) is correct because it is found by using the section formula to determine the position vector of point B.

Option C ($- \hat i + \hat j$) is correct because it represents the position vector of the midpoint C, calculated by averaging the position vectors of A and B.

The correct answer is A. In a regular hexagon with center at the origin, $\overrightarrow {\mathbf{DE}}$ is equal to $- \overrightarrow {\mathbf{AB}}$, which can be calculated from the given position vectors.

Option A (Parallelogram, which is neither a rhombus nor a rectangle) is correct because opposite sides are parallel and equal in length, but adjacent sides are not perpendicular and not all sides are equal.

Option A ($\vec 0$) is correct because the sum of the vectors from the centroid to the midpoints of the sides of a triangle is always the zero vector.

Option C ($\pm \frac{5}{3}\left( {\hat i - 7\hat j + 2\hat k} \right)$) is correct because it represents a vector of magnitude $5\sqrt{6}$ along the angle bisector of the given vectors.

Option A ($\overrightarrow {DC}$) is correct because it correctly relates the given vectors using midpoint properties in a triangle.

The correct answer is C ($\overrightarrow {\mathbf{PA}} + \overrightarrow {\mathbf{PB}} - 2\overrightarrow {\mathbf{PC}} = \vec 0$) because it directly applies the midpoint theorem in vector form.

Option D ($\frac{1}{\sqrt{74}}(4 \mathbf{i}-3 \mathbf{j}-7 \mathbf{k})$) is correct because it is the unit vector obtained by normalizing the cross product of the given vectors.

Option D (9) is correct because by substituting the line equation into the plane equation, we can find the values of c and d, and their sum is 9.

Option B (-6) is correct because it represents the value of $\lambda$ for which AB is parallel to CD, calculated using the condition for parallel vectors.

Option A (3) is correct because (a+b+c).(p+q+r) simplifies to 3 when p, q, and r are defined as given.

Option C (5 units) is correct because it represents the length of the median vector calculated using the midpoint formula.

Option A is correct because the centroid of a triangle with given position vectors is found by averaging the vectors.

Option A do not intersect for any values of $l$ and $m$ is correct because the system of equations derived from equating the two lines has no solution.

Option D (No value of ${c_1}$ can be found) is correct because the condition for coplanarity leads to a contradiction given the values of ${c_2}$ and ${c_3}$.

Option A (0) is correct because when the vectors are coplanar, their scalar triple product is zero, leading to λ = 0.

Option D ($a=9, b=3$) is correct because collinear vectors are scalar multiples of each other, leading to the ratios $\frac{3}{a}=\frac{1}{b}=\frac{-5}{-15}$.

Option C ($\sqrt {593}$) is correct because the length of the longer diagonal is calculated using the parallelogram law of vector addition and is found to be $\sqrt {593}$.

Option C (${O C}=2{b}-{a}$) is correct because using the midpoint formula, the position vector of C can be expressed as 2{b} - {a}.