Find the lengths of each of the following vectors

(a) 2i + 4j + 3k (b) 5i - 2j + k

(c) 2i - k (d) 5i

(e) 3i - 2j - k (f) i + j + k

Generally, length of vector means the magnitude of the vector.

So, given a vector

R = a•i + b•j + c•k

Then, it magnitude can be caused using

|R| = √ (a²+b²+c²)

So, applying this to each of the vector given.

(a) 2i + 4j + 3k

The length is

L = √ (2²+4²+3²)

L = √ (4+16+9)

L = √29

L = 5.385 unit

(b) 5i - 2j + k

Note that k means 1k

The length is

L = √ (5² + (-2) ²+1²)

Note that, - * - = +

L = √ (25+4+1)

L = √30

L = 5.477 unit

(c) 2i - k

Note that, since there is no component j implies that j component is 0

L = 2i + 0j - 1k

The length is

L = √ (2²+0² + (-1) ²)

L = √ (4+0+1)

L = √5

L = 2.236 unit

(d) 5i

Same as above no is j-component and k-component

L = 5i + 0j + 0k

The length is

L = √ (5²+0²+0²)

L = √ (25+0+0)

L = √25

L = 5 unit

(e) 3i - 2j - k

The length is

L = √ (3² + (-2) ² + (-1) ²)

L = √ (9+4+1)

L = √14

L = 3.742 unit

(f) i + j + k

The length is

L = √ (1²+1²+1²)

L = √ (1+1+1)

L = √3

L = 1.7321 unit