Study guide and
6 practice problems
on:
Dot product is positive for vectors in the same general direction
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If two vectors point along the same 'general direction', their dot product is positive.
That is, if the angle between two vectors is less than $\pi/2$, their dot product is positive.
All of the following pairs of vectors have positive dot product:
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Practice problems
Let $\mathbf{O}, \mathbf{N}, \mathbf{B}$ be three points as shown.
(a) What is the sign of the dot product $(\mathbf{B} - \mathbf{O}) \cdot (\mathbf{N} - \mathbf{O})$?
(b) What is the sign of the dot product $(\mathbf{B} - \mathbf{N}) \cdot (\mathbf{N} - \mathbf{O})$?
Solution
Of the unit vectors $\mathbf{A}$ through $\mathbf{H}$,
Which have a positive dot product with $\mathbf{A}$?
Which have a negative dot product with $\mathbf{A}$?
Which have zero dot product with $\mathbf{A}$?
Which has the largest dot product with $\mathbf{A}$?
Which has the most negative dot product with $\mathbf{A}$?
Solution
Show that the component (scalar projection) of $\bfa$ along $\bfb$ is positive if the angle between $\bfa$ and $\bfb$ is less than $\pi/2$. Show that it is negative if the angle is greater than $\pi/2$.
Solution
Sketch the region in 2d satisfied by $ ( 1, 1 ) \cdot \bfx \geq 2.$
Solution
Without calculation, determine if $\int_C \mathbf{F}\cdot d\mathbf{r}$ is positive, negative, or zero. $\mathbf{F}$ is the vector field pictured below. $C$ is the red circle traced out clockwise.
Solution
Without calculation or application of any theorems, determine if $\int_C \mathbf{F}\cdot d\mathbf{r}$ is positive, negative, or zero. $\mathbf{F}$ is the vector field pictured below. $C$ is the circle traced out as indicated.
Solution
