If Four Points A, B, C and D with Position Vectors 4 ^ I + 3 ^ J + 3 ^ K , 5 ^ I + X ^ J + 7 ^ K , 5 ^ I + 3 ^ J and - Mathematics

प्रश्न

If four points A, B, C and D with position vectors 4 $\hat{i} + 3$ $\hat{j} + 3$ $\hat{k}, 5$ $\hat{i} +$ $x \hat{j} + 7$ $\hat{k}, 5$ $\hat{i} + 3$ $\hat{j}$ and $7 \hat{i} + 6 \hat{j} + \hat{k}$ respectively are coplanar, then find the value of x.

बेरीज

उत्तर

Let $\vec{O A} = 4 \hat{i} + 3 \hat{j} + 3 \hat{k}, \vec{O B} = 5 \hat{i} + x \hat{j} + 7 \hat{k}, \vec{O C} = 5 \hat{i} + 3 \hat{j}$ and $\vec{O D} = 7 \hat{i} + 6 \hat{j} + \hat{k}$ .

$∴ \vec{A B} = (5 \hat{i} + x \hat{j} + 7 \hat{k}) - (4 \hat{i} + 3 \hat{j} + 3 \hat{k}) = \hat{i} + (x - 3) \hat{j} + 4 \hat{k}$

$\vec{A C} = (5 \hat{i} + 3 \hat{j}) - (4 \hat{i} + 3 \hat{j} + 3 \hat{k}) = \hat{i} - 3 \hat{k}$

$\vec{A D} = (7 \hat{i} + 6 \hat{j} + \hat{k}) - (4 \hat{i} + 3 \hat{j} + 3 \hat{k}) = 3 \hat{i} + 3 \hat{j} - 2 \hat{k}$

Since the given four points are coplanar, so the vectors $\vec{A B}, \vec{A C}$ and $\vec{A D}$ are also coplanar.

$∴ [\begin{matrix} \vec{A B} & \vec{A C} & \vec{A D} \end{matrix}] = 0$

$\Rightarrow | \begin{matrix} 1 & x - 3 & 4 \\ 1 & 0 & - 3 \\ 3 & 3 & - 2 \end{matrix} | = 0$

$\Rightarrow 1 (0 + 9) - (x - 3) (- 2 + 9) + 4 (3 - 0) = 0$

$\Rightarrow 9 - 7 x + 21 + 12 = 0$

$\Rightarrow 7 x = 42$

$\Rightarrow x = 6$

Thus, the value of x is 6.

shaalaa.com

Scalar Triple Product of Vectors

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q 14 Q 13 Q 1

पाठ 26: Scalar Triple Product - Exercise 26.1 [पृष्ठ १७]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

पाठ 26 Scalar Triple Product
Exercise 26.1 | Q 14 | पृष्ठ १७

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

view
व्हिडिओ ट्यूटोरियल For All Subjects
Scalar Triple Product of Vectors

video tutorial00:04:25

संबंधित प्रश्‍न

Show that the four points A(4, 5, 1), B(0, –1, –1), C(3, 9, 4) and D(–4, 4, 4) are coplanar.

Find λ, if the vectors $\vec{a} = \hat{i} + 3 \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} - \hat{j} - \hat{k} and \vec{c} = λ \hat{j} + 3 \hat{k}$ are coplanar.

Find the volume of the parallelopiped whose coterminus edges are given by vectors

$2 \hat{i} + 3 \hat{j} - 4 \hat{k}, 5 \hat{i} + 7 \hat{j} + 5 \hat{k} and 4 \hat{i} + 5 \hat{j} - 2 \hat{k}$

Show that the four points A, B, C and D with position vectors $4 \hat{i} + 5 \hat{j} + \hat{k}$ , $- \hat{j} - \hat{k}$ , $3 \hat{i} + 9 \hat{j} + 4 \hat{k}$ and $4 (- \hat{i} + \hat{j} + \hat{k})$ respectively are coplanar

Evaluate the following:

$[\hat{i} \hat{j} \hat{k}] + [\hat{j} \hat{k} \hat{i}] + [\hat{k} \hat{i} \hat{j}]$

Find $[\vec{a} \vec{b} \vec{c}]$ , when $\vec{a} = 2 \hat{i} - 3 \hat{j}, \vec{b} = \hat{i} + \hat{j} - \hat{k} and \vec{c} = 3 \hat{i} - \hat{k}$

Find the volume of the parallelopiped whose coterminous edges are represented by the vector:

$\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}, \vec{c} = 3 \hat{i} - \hat{j} - 2 \hat{k}$

Find the volume of the parallelopiped whose coterminous edges are represented by the vector:

$\vec{a} = 11 \hat{i}, \vec{b} = 2 \hat{j}, \vec{c} = 13 \hat{k}$

Find the value of λ so that the following vector is coplanar:

$\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - 3 \hat{k}, \vec{c} = λ \hat{i} + λ \hat{j} + 5 \hat{k}$

Find the value of λ so that the following vector is coplanar:

$\vec{a} = \hat{i} + 2 \hat{j} - 3 \hat{k}, \vec{b} = 3 \hat{i} + λ \hat{j} + \hat{k}, \vec{c} = \hat{i} + 2 \hat{j} + 2 \hat{k}$

Show that the points A (−1, 4, −3), B (3, 2, −5), C (−3, 8, −5) and D (−3, 2, 1) are coplanar.

If $\vec{a,} \vec{b}$ $are non-collinear vectors, then find the value of [\vec{a} \vec{b} \hat{i}] \hat{i} + [\vec{a} \vec{b} \hat{j}] \hat{j} + [\vec{a} \vec{b} \hat{k}] \hat{k} .$

If the vectors (sec² A) $\hat{i} + \hat{j} + \hat{k}, \hat{i} + (\sec^{2} B) \hat{j} + \hat{k}, \hat{i} + \hat{j} + (\sec^{2} C) \hat{k}$ are coplanar, then find the value of cosec² A + cosec² B + cosec² C.

If $\vec{a,} \vec{b,} \vec{c}$ are non-coplanar vectors, then find the value of $\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})}$ .

If $\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0$ for some non-zero vector $\vec{r},$ then the value of $[\vec{a} \vec{b} \vec{c}],$ is

If $\vec{a,} \vec{b,} \vec{c}$ are non-coplanar vectors, then $\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}} + \frac{\vec{b} \cdot (\vec{a} \times \vec{c})}{\vec{c} \cdot (\vec{a} \times \vec{b})}$ is equal to

Let $\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}, \vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k} and \vec{c} = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} k$ 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 $\vec{a} and \vec{b}$ is $\frac{π}{6},$ , 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

If $[2 \vec{a} + 4 \vec{b} \vec{c} \vec{d}] = λ [\vec{a} \vec{c} \vec{d}] + μ [\vec{b} \vec{c} \vec{d}],$ then λ + μ =

If the vectors $4 \hat{i} + 11 \hat{j} + m \hat{k}, 7 \hat{i} + 2 \hat{j} + 6 \hat{k} and \hat{i} + 5 \hat{j} + 4 \hat{k}$ are coplanar, then m =

If $\vec{a,} \vec{b,} \vec{c}$ are three non-coplanar vectors, then $(\vec{a} + \vec{b} + \vec{c}) . [(\vec{a} + \vec{b}) \times (\vec{a} + \vec{c})]$ equals

$(\vec{a} + 2 \vec{b} - \vec{c}) \cdot {(\vec{a} - \vec{b}) \times (\vec{a} - \vec{b} - \vec{c})}$ is equal to

Find the value of p, if the vectors $\hat{i} - 2 \hat{j} + \hat{k}, 2 \hat{i} - 5 \hat{j} + p \hat{k}, 5 \hat{i} - 9 \hat{j} + 4 \hat{k}$ are coplanar.

Determine where $\bar{a}$ and $\bar{b}$ are orthogonal, parallel or neither.

$\bar{a} = - \frac{3}{5} \hat{i} + \frac{1}{2} \hat{j} + \frac{1}{3} \hat{k}, \bar{b} = 5 \hat{i} + 4 \hat{j} + 3 \hat{k}$

If a vector has direction angles 45° and 60°, find the third direction angle.

If $\vec{a} = \hat{i} - 2 \hat{j} + 3 \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - 2 \hat{k}, \vec{c} = 3 \hat{i} + 2 \hat{j} + \hat{k}$ , find $\vec{a} \cdot (\vec{b} \times \vec{c})$

Determine whether the three vectors $2 \hat{i} + 3 \hat{j} + \hat{k}, \hat{i} - 2 \hat{j} + 2 \hat{k}$ and $3 \hat{i} + \hat{j} + 3 \hat{k}$ are coplanar

Let $\vec{a}, \vec{b}, \vec{c}$ 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 $\vec{b}$ is $\frac{π}{6}$ , show that ${[\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]}^{2} = \frac{1}{4} {| \vec{a} |}^{2} {| \vec{b} |}^{2}$

If θ is the angle between the unit vectors $\bar{a}$ and $\bar{b}$ , the $\cos θ = \frac{θ}{2}$ = ______.

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors, then the value of $\frac{\vec{a} . (\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) . \vec{b}} + \frac{\vec{b} . (\vec{a} \times \vec{c})}{\vec{c} . (\vec{a} \times \vec{b})}$ is ______.

Find the volume of the parallelopiped whose coterminous edges are $2 \hat{i} - 3 \hat{j}, \hat{i} + \hat{j} - \hat{k}$ and $3 \hat{i} - \hat{k}$ .

If $\bar{u} = \hat{i} - 2 \hat{j} + \hat{k}, \bar{v} = 3 \hat{i} + \hat{k}$ and $\bar{w} = \hat{j} - \hat{k}$ are given vectors, then find $[\bar{u} \times \bar{v} \bar{u} \times \bar{w} \bar{v} \times \bar{w}]$

Determine whether $\bar{a} and \bar{b}$ is orthogonal, parallel or neither.

$\bar{a} = - \frac{3}{5} \hat{i} + \frac{1}{2} \hat{j} + \frac{1}{3} \hat{k}, \bar{b} = 5 \hat{i} + 4 \hat{j} + 3 \hat{k}$

If $u = \hat{i} - 2 \hat{j} + \hat{k}, \bar{r} = 3 \hat{i} + \hat{k} and w = \hat{j}, \hat{k}$ are given vectors, then find $[\bar{u} + \bar{w}] . [(\bar{w} \times \bar{r}) \times (\bar{r} \times \bar{w})]$

If $\bar{u} = \hat{i} - 2 \hat{j} + \hat{k}, \bar{v} = 3 \hat{i} + \hat{k} and \bar{w} = \hat{j} - \hat{k}$ are given vectors, then find $[\bar{u} + \bar{w}] \cdot [(\bar{u} \times \bar{v}) \times (\bar{v} \times \bar{w})]$

Find the volume of a tetrahedron whose vertices are A(−1, 2, 3) B(3, −2, 1), C(2, 1, 3) and D(−1, −2, 4).

If Four Points A, B, C and D with Position Vectors 4 ^ I + 3 ^ J + 3 ^ K , 5 ^ I + X ^ J + 7 ^ K , 5 ^ I + 3 ^ J and - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Select a course

If Four Points A, B, C and D with Position Vectors 4 ^ I + 3 ^ J + 3 ^ K , 5 ^ I + X ^ J + 7 ^ K , 5 ^ I + 3 ^ J and - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

व्हिडिओ ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्‍न

Select a course

उत्तर