Name: Proof of Inequalities Using Dot Product
Uploaded: 2019-07-19
Duration: 13 min 9 s
Description: Now we know the dot product of vectors. So let us learn how to use the dot product of vectors to prove standard inequalities in this video

Question

If

\vec{a}, \vec{b}, \vec{c}

are non-coplanar find the point of intersection of the line passing through the points

2 \vec{a} + 3 \vec{b} - \vec{c}, 3 \vec{a} + 4 \vec{b} - 2 \vec{c}

with the line joining the points

\vec{a} - 2 \vec{b} + 3 \vec{c}, \vec{a} - 6 \vec{b} + 6 \vec{c}

.

Accepted Answer

Let, $\vec{O A} = 2 \vec{a} + 3 \vec{b} - \vec{c}, \vec{O B} = 3 \vec{a} + 4 \vec{b} - 2 \vec{c}, \vec{O C} = \vec{a} - 2 \vec{b} + 3 \vec{c}$ and $\vec{O A} = \vec{a} - 6 \vec{b} + 6 \vec{c}$

The vector equation of the line joining the points $\vec{O A}, \vec{O B}$ is $\vec{r} = \vec{O A} + t (\vec{O B} - \vec{O A}), t \in R$

$= 2 \vec{a} + 3 \vec{b} - \vec{c} + t (3 \vec{a} + 4 \vec{b} - 2 \vec{c} - 2 \vec{a} - 3 \vec{b} + \vec{c})$

$\vec{r} = 2 \vec{a} + 3 \vec{b} - \vec{c} + t (\vec{a} + \vec{b} - \vec{c})$ ...(i)

The vector equation of the line joining the points $\vec{O C}, \vec{O D}$ is $\vec{r} = \vec{O C} + s (\vec{O D} - \vec{O C}), s \in R$

$= \vec{a} - 2 \vec{b} + 3 \vec{c} + s (\vec{a} - 6 \vec{b} + 6 \vec{c} - \vec{a} + 2 \vec{b} - 3 \vec{c})$

$\vec{r} = \vec{a} - 2 \vec{b} + 3 \vec{c} + s (- 4 \vec{b} + 3 \vec{c})$

$= \vec{a} + (- 2 - 4 s) \vec{b} + (3 + 3 s) \vec{c}$ ...(ii)

Comparing $\vec{a}, \vec{b}, \vec{c}$ coefficients of (i) and (ii), we get

$2 + t = 1$

$\Rightarrow t = - 1$

And $3 + t = - 2 - 4 s$

$\Rightarrow 2 = - 2 + 4 s$

$\Rightarrow s = - 1$

Substitute $t$ in (i),

$\vec{r} = 2 \vec{a} + 3 \vec{b} - \vec{c} + (- 1) (\vec{a} + \vec{b} - \vec{c})$

$\vec{r} = \vec{a} + 2 \vec{b}$

$∴$ The vector equation is $\bar{r} = \vec{a} + 2 \vec{b}$ .

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar find the point of intersection of the line passing through the points $2 \vec{a} + 3 \vec{b} - \vec{c}, 3 \vec{a} + 4 \vec{b} - 2 \vec{c}$ with the line joining the points $\vec{a} - 2 \vec{b} + 3 \vec{c}, \vec{a} - 6 \vec{b} + 6 \vec{c}$ .

Important Questions on Addition of Vectors

Learn and Prepare for any exam you want !

If a→, b→, c→ are non-coplanar find the point of intersection of the line passing through the points 2a→+3b→-c→, 3a→+4b→-2c→ with the line joining the points a→-2b→+3c→, a→-6b→+6c→.

Important Questions on Addition of Vectors

Learn and Prepare for any exam you want !

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar find the point of intersection of the line passing through the points $2 \vec{a} + 3 \vec{b} - \vec{c}, 3 \vec{a} + 4 \vec{b} - 2 \vec{c}$ with the line joining the points $\vec{a} - 2 \vec{b} + 3 \vec{c}, \vec{a} - 6 \vec{b} + 6 \vec{c}$ .