Question Type 1: Finding the dot product of two different vectors Exercises

Compute the dot product $a\\cdot (b + 2c)$ for $a=(1,0,1)$, $b=(3,2,1)$ and $c=(5,6,1)$.

Calculate $b\cdot c - 2\,a\cdot c$ for $a=(1,0,1)$, $b=(3,2,1)$ and $c=(5,6,1)$.

Determine whether the vectors $a=(1,0,1)$ and $c=(5,6,1)$ are perpendicular.

Compute $b\cdot (a - c)$ for $a=(1,0,1)$, $b=(3,2,1)$ and $c=(5,6,1)$.

Find the angle between the vectors $p=(3,-1,2)$ and $q=(1,2,-2)$.

Find the angle between the vectors $u=(1,4,3)$ and $v=(2,1,2)$.

Two unit vectors $u$ and $v$ satisfy $u\cdot v = \tfrac12$. Find the angle between $u$ and $v$.

Determine the value of $k$ such that the vectors $(k,2,3)$ and $(1,k,4)$ are perpendicular.

Verify the distributive property $a\cdot(b + 2c) = a\cdot b + 2\,a\cdot c$ for $a=(1,1,0)$, $b=(2,-1,3)$ and $c=(0,1,2)$ and compute the resulting value.

Show that the angle between the vectors $(1,1,1)$ and $(1,0,0)$ is acute and find its measure.

Find all real values of $t$ such that the vectors $(1,t,t^2)$ and $(2,-1,1)$ are orthogonal.

In a parallelogram, the diagonals are given by $a + b$ and $a - b$ for $a=(2,1,0)$ and $b=(1,3,4)$. Find the angle between the diagonals.