If the vertices of a triangle are the points with position vectors a1i-a2j-a3k- -160-b1i-b2j-b3k- -160-c1i-c2j-c3k- what are the vectors determined by its sides- Find the length of these vectors-

Given the vertices of a triangle A, B and C with position vectors nbsp;a1i^ #160;+ #160;a2j^ #160;+ #160;a3k^, #160;b1i^+b2j^+b3k^ nbsp;and nbsp ...

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Byju's Answer

Standard VII

Mathematics

Perpendicular Bisector and It's Construction

If the vertic...

Question

If the vertices of a triangle are the points with position vectors $a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}, b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}, c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k},$ what are the vectors determined by its sides? Find the length of these vectors.

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Solution

Given the vertices of a triangle A, B and C with position vectors $a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}, b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$ and $c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}$ respectively. Then,
$\vec{A B} = (b_{1} - a_{1}) \hat{i} + (b_{2} - a_{2}) \hat{j} + (b_{3} - a_{3}) \hat{k} . \vec{B C} = (c_{1} - b_{1}) \hat{i} + (c_{2} - b_{2}) \hat{j} + (b_{3} - a_{3}) \hat{k} . \vec{C A} = (a_{1} - c_{1}) \hat{i} + (a_{2} - c_{2}) \hat{j} + (a_{3} - c_{3}) \hat{k} .$
Therefore, the length of these vectors are:

$|\vec{A B}| = \sqrt{(b_{1} - a_{1})^{2} + (b_{2} - a_{2})^{2} + (b_{3} - a_{3})^{2}} . |\vec{B C}| = \sqrt{(c_{1} - b_{1})^{2} + (c_{2} - b_{2})^{2} + (c_{3} - b_{3})^{2}} . |\vec{C A}| = \sqrt{(a_{1} - c_{1})^{2} + (a_{2} - c_{2})^{2} + (a_{3} - c_{3})^{2}} .$

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Similar questions

Q. If

\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}, \vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k} and \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k},

then verify that

\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} .

Q. Let

a = a_{1} i + a_{2} j + a_{3} k, b = b_{1} i + b_{2} j + b_{3} k,

and

c = c_{1} i + c_{2} j + c_{3} k

be three non-zero vectors such that

c

is unit vector perpendicular to both vectors

a

and

b

. If the angle between vectors

a

and

b

\frac{π}{6}

, then

{∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2}

is equal to

Q. Let

a = a_{1} i + a_{2} j + a_{3} k, b = b_{1} i + b_{2} j + b_{3} k

and

c = c_{1} i + c_{2} j + c_{3} k

be three non-zero such that

c

is a unit perpendicular to both vectors

a

and

b

. If the angle between vectors

a

and

b

\frac{π}{6},

then

{∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2}

is equal to

Q. Let

a = a_{1} i + a_{2} j + a_{3} k, b = b_{1} i + b_{2} j + b_{3} k

and

c = c_{1} i + c_{2} j + c_{3} k

be three non-zero vectors such that

c

is a unit vector perpendicular to both

a

and

b

. If the angle between

a

and

b

π / 6

, then

{∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣}^{2}

is equal to

Q. Let

\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}, \vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k} and \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}

be three non-zero vectors such that

\vec{c}

is a unit vector perpendicular to both

\vec{a} and \vec{b}

. If the angle between

\vec{a} and \vec{b}

\frac{π}{6},

then

{|\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix}|}^{2}

is equal to

(a) 0
(b) 1
(c)

\frac{1}{4} {|\vec{a}|}^{2} {|\vec{b}|}^{2}

(d)

\frac{3}{4} {|\vec{a}|}^{2} {|\vec{b}|}^{2}

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If the vertices of a triangle are the points with position vectors a1i^+a2j^+a3k^, b1i^+b2j^+b3k^, c1i^+c2j^+c3k^, what are the vectors determined by its sides? Find the length of these vectors.

If the vertices of a triangle are the points with position vectors $a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}, b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}, c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k},$ what are the vectors determined by its sides? Find the length of these vectors.