Question

For the three vectors a=(1,1,1), b=(5,2,-6) and c=(3,-7,5) show that a∙b=a∙c and interpret the results.

Answer 1

Answer on Question#52769 - Math - Other

For the three vectors $a = (1,1,1)$, $b = (5,2,-6)$ and $c = (3,-7,5)$ show that $a \cdot b = a \cdot c$ and interpret the results.

Solution:

The dot product is given by

$$A \cdot B = A _ {x} B _ {x} + A _ {y} B _ {y} + A _ {z} B _ {z}$$

Therefore,

$$a \cdot b = 1 \cdot 5 + 1 \cdot 2 + 1 \cdot (- 6) = 1$$ $$a \cdot c = 1 \cdot 3 + 1 \cdot (- 7) + 1 \cdot 5 = 1$$

Also the dot product can be written in the following way

$$A \cdot B = | A | \cdot | B | \cos \varphi$$

$|B|$ cos $\varphi$ gives the projection of vector $B$ on the vector $A$ ($\varphi$ – is the angle between vectors $A$ and $B$). Therefore, vectors $b$ and $c$ have the same projection on the vector $a$.

**Answer**: vectors $b$ and $c$ have the same projection on the vector $a$.

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