Question

$Given \: two \: vectors \: \overrightarrow{\mathrm{A}} = 2{\hat{i}+4\hat{j}} + 5k \: \: and \: \overrightarrow{\mathrm{B}} = {\hat{i}+\hat{j}} + 6k. \: Find \: the \: product \: \overrightarrow{\mathrm{A}}.\overrightarrow{\mathrm{B}} \: and \: \overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}. \: what \: is \: the \: angle \: between \: them.$


★ Trusted answers required. Pls​

Answer 1

Answer:

♦  A . B = 36

♦  A × B = 19î - 7j - 2k

♦  Angle between them is approximately 29°

Step-by-step explanation:

Given:

Two vectors:

• A = 2i + 4j + 5k
• B = i + j + 6k

To find:

• A . B
• A × B
• Angle between vector A and vector B

Solution:

For finding the dot product (scalar product);

Just multiply the numbers with their corresponding i, j, k.

   A . B

= (2i + 4j + 5k) . (i + j + 6k)

= [2(1)] + [4(1)] + [5(6)]

= 2 + 4 + 30

= 36

Now, for finding the cross product (vector product);

consider

★ A vector to be (aî + bj + ck) and

★ B vector to be (xî + yj + zk)

So,

• a = 2
• b = 4
• c = 5
• x = 1
• y = 1
• z = 6

   A × B

= (bz - cy)î + (cx - az)j + (ay - bx)k

= (24 - 5)î + (5 - 12)j + (2 - 4)k

= 19î - 7j - 2k

For finding the angle between them:

$\sf{cos \theta = \dfrac{ \overrightarrow{a}. \overrightarrow{b}}{|(a)||(b)|} }$

$\implies \sf{cos \theta = \dfrac{ 36 }{( \sqrt{ {2}^{2} + {4}^{2} + {5}^{2} } ) \times ( \sqrt{ {1}^{2} + {1}^{2} + {6}^{2} } ) } }$

$\implies \sf{ \cos \theta = \dfrac{ 36 }{ \sqrt{45} \times \sqrt{38} } }$

$\implies \sf{ \theta = cos^{ - 1} \bigg( \dfrac{ 36 }{ \sqrt{45} \times \sqrt{38} } \bigg) }$

$\implies \boxed{ \bf{ \theta \approx 29^{ \circ} } }$

Answer 2

Answer:

Given

a= 2i+3j+4j

b=3i+4j+5k

The scalar product of these two vectors is-

a.b= |a| |b| cosθ

Where θ is the angle between the two vectors.

(2i+3j+4j)(3i+4j+5k) =

|a| |b| cosθ

Or,

6+12+20 = |a| |b| cosθ

Where,

|a| =sqrt(2^2+3^2+4^2)

|b|= sqrt(3^2+4^2+5^2)

Therefore,

38= sqrt(29).sqrt(50)cosθ

cosθ= 38/sqrt(1450)

cosθ=38/38.07

cosθ= 0.998

θ= 1.54 degree.

The angle between the two vectors will be about 1.54 degree.