Riigieksamid (Estonian State Exams) · Mathematics

Analytic Geometry and Vectors

10 practice questions, each with the answer and an explanation.

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All 10 questions

Given 3D vectors $u = (1, 2, - 1)$ and $v = (3, - 1, 2)$ , what is their dot product $u \cdot v$ ?
Find the cross product $a \times b$ for vectors $a = (1, 0, 0)$ and $b = (0, 1, 0)$ in the standard Cartesian coordinate system.
What is the cosine of the angle $θ$ between vectors $p = (2, 2, 1)$ and $q = (1, 2, 2)$ ?
What is the perpendicular distance from point $P (1, - 2)$ to the straight line $3 x - 4 y + 5 = 0$ in the coordinate plane?
Identify the centre and radius of the sphere given by $x^{2} + y^{2} + z^{2} - 4 x + 6 y - 2 z - 11 = 0$ .
Determine the value of $λ$ such that the vectors $u = (2, - 1, 3)$ and $v = (λ, 4, - 2)$ are orthogonal.
What is the equation of the plane passing through the point $P (1, 2, 3)$ with normal vector $n = (2, - 1, 4)$ ?
What is the volume of a right circular cone with base radius $r = 3$ cm and slant height $s = 5$ cm?
Find the eccentricity of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ .
What is the acute angle of intersection between the two straight lines $y = 2 x + 1$ and $y = 3 x - 4$ ?

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