Scalar and vector products of two vectors. Given two vectors 𝐚=3 𝐱̂+4 ŷ-5 𝐳̂ and 𝐛=-𝐱̂+2 𝐲̂+6 𝐳̂

step 1 Okay in the first part we have to find the length in a part we have to find the length of body s

Scalar and vector products of two vectors. Given two vectors...

Scalar and vector products of two vectors. Given two vectors $\mathbf{a}=3 \hat{\mathbf{x}}+4 \hat{y}-5 \hat{\mathbf{z}}$ and $\mathbf{b}=-\hat{\mathbf{x}}+2 \hat{\mathbf{y}}+6 \hat{\mathbf{z}}$, calculate by vector methods: (a) The length of each Ans, $a=\sqrt{50} ; b=\sqrt{41}$. (b) The scalar product a \cdot b Ans. $-25$. (c) The included angle between them Ans. $123.5^{\circ}$. (d) The direction cosines for each (e) The vector sum and difference $a+b$ and $a-b$

Scalar and vector products of two vectors. Given two vectors $\mathbf{a}=3 \hat{\mathbf{x}}+4 \hat{y}-5 \hat{\mathbf{z}}$ and $\mathbf{b}=-\hat{\mathbf{x}}+2 \hat{\mathbf{y}}+6 \hat{\mathbf{z}}$, calculate by
vector methods:
(a) The length of each Ans, $a=\sqrt{50} ; b=\sqrt{41}$.
(b) The scalar product a \cdot b Ans. $-25$.
(c) The included angle between them Ans. $123.5^{\circ}$.
(d) The direction cosines for each
(e) The vector sum and difference $a+b$ and $a-b$

Key Concepts

Direction Cosines

Direction cosines are the cosines of the angles between a vector and each of the coordinate axes. They are obtained by dividing each component of the vector by its magnitude. This provides a standardized way to describe the orientation of a vector in space, independent of its magnitude.

Included Angle between Vectors

The included angle between two vectors is the angle formed by them when they are placed tail-to-tail. It can be determined by rearranging the dot product formula, where the cosine of the angle is equal to the dot product divided by the product of the magnitudes of the vectors. This concept is key in understanding the geometric relationship between vectors.

Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors by adding or subtracting their corresponding components. These operations obey the parallelogram law and are essential for determining resultant vectors in physics and engineering problems. They allow for the resolution and recombination of forces and other vector quantities in a coherent manner.

Vector Magnitude

The vector magnitude, also known as the norm of the vector, is a measure of its length. It is calculated as the square root of the sum of the squares of its components. This concept is fundamental in vector algebra because it provides a way to quantify the size of a vector regardless of its direction.

Dot Product

The dot product (or scalar product) of two vectors is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. It is computed by multiplying corresponding components of the vectors and then summing those products. The result can also be interpreted as the product of the magnitudes of the two vectors and the cosine of the angle between them, making it useful for determining the angle between vectors.

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