Find the Area of the Parallelogram Determined by the Vector 3 ^ I + ^ J − 2 ^ K and ^ I − 3 ^ J + 4 ^ K - Mathematics

प्रश्न

Find the area of the parallelogram determined by the vector $3 \hat{i} + \hat{j} - 2 \hat{k} and \hat{i} - 3 \hat{j} + 4 \hat{k}$ .

योग

उत्तर

$Let :$
$\vec{a} = 3 \hat{i} + \hat{j} - 2 \hat{k}$
$\vec{b} = 1 \hat{i} - 3 \hat{j} + 4 \hat{k}$
$\vec{a} \times \vec{b} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 1 & - 2 \\ 1 & - 3 & 4 \end{matrix} |$
$= \hat{i} (4 - 6) - \hat{j} (12 + 2) + \hat{k} (- 9 - 1)$
$= - 2 \hat{i} - 14 \hat{j} - 10 \hat{k}$
$Area of the parallelogram = | \vec{a} \times \vec{b} |$
$= \sqrt{{(- 2)}^{2} + {(- 14)}^{2} + {(- 10)}^{2}}$
$= \sqrt{300}$
$= 10 \sqrt{3} sq. units$

shaalaa.com

Product of Two Vectors - Vector (Or Cross) Product of Two Vectors

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 8.3 Q 8.2 Q 8.4

अध्याय 25: Vector or Cross Product - Exercise 25.1 [पृष्ठ २९]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12

अध्याय 25 Vector or Cross Product
Exercise 25.1 | Q 8.3 | पृष्ठ २९

वीडियो ट्यूटोरियलVIEW ALL [2]

view
वीडियो ट्यूटोरियल For All Subjects
Product of Two Vectors - Vector (Or Cross) Product of Two Vectors

video tutorial01:10:14
Product of Two Vectors - Vector (Or Cross) Product of Two Vectors

video tutorial01:04:29

संबंधित प्रश्न

If a unit vector $\vec{a}$ makes an angles $\frac{π}{3}$ with $\hat{i}, \frac{π}{4}$ with $\hat{j}$ and an acute angle θ with $\hat{k}$ , then find θ and, hence the compounds of $\vec{a}$ .

Show that $(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) = 2 (\vec{a} \times \vec{b})$ .

Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

If A, B, C are three non- collinear points with position vectors $\vec{a}, \vec{b}, \vec{c}$ , respectively, then show that the length of the perpendicular from Con AB is $\frac{| (\vec{a} \times \vec{b}) + (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{a}) |}{| (\vec{b} - \vec{a}) |}$

$If \vec{a} = \hat{i} + 3 \hat{j} - 2 \hat{k} and \vec{b} = - \hat{i} + 3 \hat{k}, find | \vec{a} \times \vec{b} | .$

If $\vec{a} = 3 \hat{i} + 4 \hat{j} and \vec{b} = \hat{i} + \hat{j} + \hat{k},$ find the value of $| \vec{a} \times \vec{b} | .$

Find a unit vector perpendicular to both the vectors $4 \hat{i} - \hat{j} + 3 \hat{k} and - 2 \hat{i} + \hat{j} - 2 \hat{k} .$

Find the area of the parallelogram determined by the vector $2 \hat{i} and 3 \hat{j}$ .

Given $\vec{a} = \frac{1}{7} (2 \hat{i} + 3 \hat{j} + 6 \hat{k}), \vec{b} = \frac{1}{7} (3 \hat{i} - 6 \hat{j} + 2 \hat{k}), \vec{c} = \frac{1}{7} (6 \hat{i} + 2 \hat{j} - 3 \hat{k}), \hat{i}, \hat{j}, \hat{k}$ being a right handed orthogonal system of unit vectors in space, show that $\vec{a}, \vec{b}, \vec{c}$ is also another system.

If | \vec{a} | = 13, | \vec{b} | = 5 and \vec{a} . \vec{b} = 60, then find | \vec{a} \times \vec{b} | .

If $\vec{a,} \vec{b,} \vec{c}$ are three unit vectors such that $\vec{a} \times \vec{b} = \vec{c}, \vec{b} \times \vec{c} = \vec{a,} \vec{c} \times \vec{a} = \vec{b} .$ Show that $\vec{a,} \vec{b,} \vec{c}$ form an orthonormal right handed triad of unit vectors.

Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B and Care A (3, −1, 2), B (1, −1, −3) and C (4, −3, 1).

if $\vec{a} = \hat{i} - 2 \hat{j} + 3 \hat{k}, and \vec{b} = 2 \hat{i} + 3 \hat{j} - 5 \hat{k},$ then find $\vec{a} \times \vec{b} .$ Verify th at $\vec{a} and \vec{a} \times \vec{b}$ are perpendicular to each other.

If $\vec{p} and \vec{q}$ are unit vectors forming an angle of 30°; find the area of the parallelogram having $\vec{a} = \vec{p} + 2 \vec{q} and \vec{b} = 2 \vec{p} + \vec{q}$ as its diagonals.

Define $\vec{a} \times \vec{b}$ and prove that $| \vec{a} \times \vec{b} | = (\vec{a} . \vec{b})$ tan θ, where θ is the angle between $\vec{a} and \vec{b}$ .

Let $\vec{a} = \hat{i} + 4 \hat{j} + 2 \hat{k}, \vec{b} = 3 \hat{i} - 2 \hat{j} + 7 \hat{k} and \vec{c} = 2 \hat{i} - \hat{j} + 4 \hat{k} .$ Find a vector $\vec{d}$ which is perpendicular to both $\vec{a} and \vec{d}$ $and \vec{c} \cdot \vec{d} = 15 .$

If either $\vec{a} = \vec{0} or \vec{b} = \vec{0}, then \vec{a} \times \vec{b} = \vec{0} .$ Is the converse true? Justify your answer with an example.

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

Using vectors, find the area of the triangle with vertice A(1, 2, 3), B(2, −1, 4) and C(4, 5, −1) .

Find all vectors of magnitude $10 \sqrt{3}$ that are perpendicular to the plane of $\hat{i} + 2 \hat{j} + \hat{k}$ and $- \hat{i} + 3 \hat{j} + 4 \hat{k}$ .

Write the value of $\hat{i} . (\hat{j} \times \hat{k}) + \hat{j} . (\hat{k} \times \hat{i}) + \hat{k} . (\hat{i} \times \hat{j}) .$

Write the expression for the area of the parallelogram having $\vec{a} and \vec{b}$ as its diagonals.

If $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k},$ then write the value of ${| \vec{r} \times \hat{i} |}^{2} .$

If $\vec{a} and \vec{b}$ are two vectors such that $| \vec{a} . \vec{b} | = | \vec{a} \times \vec{b} |,$ write the angle between $\vec{a} and \vec{b} .$

The unit vector perpendicular to the plane passing through points $P (\hat{i} - \hat{j} + 2 \hat{k}), Q (2 \hat{i} - \hat{k}) and R (2 \hat{j} + \hat{k})$ is

If $\vec{a} = 2 \hat{i} - 3 \hat{j} - \hat{k} and \vec{b} = \hat{i} + 4 \hat{j} - 2 \hat{k}, then \vec{a} \times \vec{b}$ is

If $| \vec{a} \times \vec{b} | = 4, | \vec{a} \cdot \vec{b} | = 2, then {| \vec{a} |}^{2} {| \vec{b} |}^{2} =$

The value of $\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}),$ is

If θ is the angle between any two vectors $\bar{a}$ and $\bar{b}$ and $| \bar{a} \cdot \bar{b} | = | \bar{a} \times \bar{b} |$ then θ is equal to ______.

(a) If $\vec{a} = \hat{i} - 2 j + 3 \vec{k}, \vec{b} = 2 \hat{i} + 3 \hat{j} - 5 \hat{k},$ prove that $\vec{a} and \vec{a} \times \vec{b}$ are perpendicular.

Find a unit vector perpendicular to both the vectors $\vec{a} and \vec{b}$ , where $\vec{a} = \hat{i} - 7 \hat{j} + 7 \hat{k}$ and $\vec{b} = 3 \hat{i} - 2 \hat{j} + 2 \hat{k}$ .

The value of λ for which the two vectors $2 \hat{i} - \hat{j} + 2 \hat{k}$ and $3 \hat{i} + λ \hat{j} + \hat{k}$ are perpendicular is ______.

What is the sum of vector $\vec{a} = \hat{i} - 2 \hat{i} + \hat{k}, \vec{b} = - 2 \hat{i} + 4 \hat{j} + 5 \hat{k}$ and $\vec{c} = \hat{i} - 6 \hat{j} - 7 \hat{k}$

Let $\vec{a}$ and $\vec{b}$ be two unit vectors and θ is the angle between them, Then $\vec{a} + \vec{b}$ is a unit vector if-

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$ satisfies. $\vec{a} \times {(\vec{r} - \vec{b}) \times \vec{a}} + \vec{b} \times {(\vec{r} - \vec{c}) \times \vec{b}} + \vec{c} \times {(\vec{r} - \vec{a}) \times \vec{c}} = \vec{0}$ , then $\vec{r}$ is equal to ______.

If $| \vec{a} \times \vec{b} | = \sqrt{3}$ and $\vec{a} . \vec{b}$ = – 3, then angle between $\vec{a}$ and $\vec{b}$ is ______.

If $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ then find a unit vector perpendicular to both $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$ .

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $| \vec{a} \times \vec{b} | = \vec{a} . \vec{b}$ , find the angle between $\vec{a}$ and $\vec{b}$ .

If $\vec{a}$ is a unit vector perpendicular to $\vec{b}$ and $(\vec{a} + 2 \vec{b}) . (3 \vec{a} - \vec{b}) = - 5$ , find $| \vec{b} |$ .

Find the Area of the Parallelogram Determined by the Vector 3 ^ I + ^ J − 2 ^ K and ^ I − 3 ^ J + 4 ^ K - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [2]

संबंधित प्रश्न

Select a course

Find the Area of the Parallelogram Determined by the Vector 3 ^ I + ^ J − 2 ^ K and ^ I − 3 ^ J + 4 ^ K - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [2]

संबंधित प्रश्न

Select a course

उत्तर