The values of k for which the points $$A(1$$,$$0$$,$$3)$$,$$B($$−$$1$$,$$3$$,$$4)$$,$$C(1$$,$$2$$,$$1)$$ and $$D(k.2$$,$$5)$$ are coplanar, are

Question

The values of k for which the points $$A(1$$,$$0$$,$$3)$$,$$B($$−$$1$$,$$3$$,$$4)$$,$$C(1$$,$$2$$,$$1)$$ and $$D(k.2$$,$$5)$$ are coplanar, are

Infinity Learn · Accepted Answer

Let $$a=(1$$,$$0$$,$$3)$$,$$b=($$−$$1$$,$$3$$,$$4)$$,$$c=(1$$,$$2$$,$$1)$$ and $$d=(k$$,$$2$$,$$5)$$ . Since A,B,C and D are coplanar, we have [dbc]$$+$$[dca]$$+$$[dab]$$=$$[abc]⇒k−$$11$$   $$2$$  $$3$$  $$2$$  $$541+k$$  $$1$$  $$1$$  $$220$$  $$5$$ $$1$$ $$3+k25103-134=1$$  −$$11$$   $$0$$    $$3$$    $$2$$    $$3$$    $$4$$     $$1$$⇒   −$$5k$$−$$15+6k$$−$$14$$−$$9k+1=$$ −$$20$$⇒   −$$8k$$−$$28=$$−$$20$$           ⇒      $$k=$$−$$1$$

$The values of k for which the points A (1, 0, 3), B (- 1, 3, 4), C (1, 2, 1) and D (k .2, 5) are coplanar, are$

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The values of k for which the points A(1,0,3),B(−1,3,4),C(1,2,1) and D(k.2,5) are coplanar, are

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$The values of k for which the points A (1, 0, 3), B (- 1, 3, 4), C (1, 2, 1) and D (k .2, 5) are coplanar, are$